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A Complete Resource Book in Mathematics for JEE Main

Dinesh Khattar

Chapter 10

Sequence and Series - all with Video Answers

Educators

+ 1 more educators

Chapter Questions

02:20

Problem 1

If $a, b, c$ are positive numbers in A.P. such that their product is 64 , then the minimum value of $b$
$(\mathrm{A})=2$
(B) $=4$
$(\mathrm{C})=1$
(D) Does not exist

Wendi Zhao
Wendi Zhao
Numerade Educator
03:09

Problem 2

If three successive terms of a G.P. with common ratio $r(r>1)$ form the sides of a $\triangle A B C$ and $[r]$ denotes greatest integer function, then $[r]+[-r]=$
(A) 0
(B) 1
(C) $-1$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
02:50

Problem 3

If $\sum_{j=1}^{21} a_{j}=693$, where $a_{1}, a_{2}, \ldots, a_{21}$, are in A.P., then $\sum_{i=0}^{10} a_{2 i+1}$ is
(A) 361
(B) 396
(C) 363
(D) data insufficient

Wendi Zhao
Wendi Zhao
Numerade Educator
02:46

Problem 4

Number of increasing geometrical progression(s) with first term unity, such that any three consecutive terms, on doubling the middle become an A.P, is
(A) 0
(B)
(C) 2
(D) infinity

Wendi Zhao
Wendi Zhao
Numerade Educator
02:20

Problem 5

If $a_{1}, a_{2}, a_{3}$ (with $\left.a_{1}>0\right)$ are in G.P. with common ratio $r$, then the value of $r$ for which the inequality $9 a_{1}+5 a_{3}$ $>14 a_{2}$ holds, cannot be in the interval
(A) $\left[1, \frac{9}{2}\right]$
(B) $(-\infty, 0)$
(C) $\left[\frac{5}{9}, 1\right]$
(D) $\left[1, \frac{9}{5}\right]$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:05

Problem 6

Let $S_{n}(1 \leq n \leq 9)$ denotes the sum of $n$ terms of series $1+22+333+\ldots+999999999$, then for $2 \leq n \leq 9$
(A) $S_{n}-S_{n-1}=\frac{1}{9}\left(10^{n}-n^{2}+n\right)$
(B) $S_{n}=\frac{1}{9}\left(10^{n}-n^{2}+2 n-2\right)$
(C) $9\left(S_{n}-S_{n-1}\right)=n\left(10^{n}-1\right)$
(D) None of these

mp
Manik Pulyani
Numerade Educator
02:27

Problem 7

If $\log _{\sqrt{5}} x+\log _{5^{n}} x+\log _{5^{4}} x+\ldots$ upto 7 terms $=35$,
then $x$ is equal to
(A) 5
(B) 25
(C) 125
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
01:27

Problem 8

If $\sum_{n=1}^{\infty} x^{n-1}=a$ and $\sum_{n=1}^{\infty} y^{n-1}=b$ where $|x|,|y|<1$,
then $\sum_{n=1}^{\infty}(x y)^{n-1}=$
(A) $a b$
(B) $\frac{a+b-1}{a b}$
(C) $\frac{1}{1-a b}$
(D) $\frac{a b}{a+b-1}$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:08

Problem 9

Let $p, q, r \in R^{+}$and $27 p q r \geq(p+q+r)^{3}$ and $3 p+4 q$ $+5 r=12$ then $p^{3}+q^{4}+r^{5}$ is equal to
(A) 3
(B) 6
(C) 2
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
03:51

Problem 10

The sum of the series $\frac{1}{1+1^{2}+1^{4}}+\frac{2}{1+2^{2}+2^{4}}+\frac{3}{1+3^{2}+3^{4}}+\ldots$ to $n$ terms
is
(A) $\frac{n\left(n^{2}+1\right)}{n^{2}+n+1}$
(B) $\frac{n(n+1)}{2\left(n^{2}+n+1\right)}$
(C) $\frac{n\left(n^{2}-1\right)}{2\left(n^{2}+n+1\right)}$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
02:23

Problem 11

$a, b, c$ are three distinct real numbers, which are in G.P. and $a+b+c=x b$. Then
(A) $x<-1$ or $x>3$
(B) $-1<x<3$
(C) $-1<x<2$
(D) $0<x<1$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:35

Problem 12

The sum of the first hundred terms of an A.P. is $x$ and the sum of the hundred terms starting from the third term is $y$. Then the common difference is
(A) $\frac{y-x}{2}$
(B) $\frac{y-x}{50}$
(C) $\frac{y-x}{100}$
(D) $\frac{y-x}{200}$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:13

Problem 13

If $\lambda=\sum_{i=1}^{\infty} \frac{1}{i^{4}}$, then $\sum_{i=1}^{\infty} \frac{1}{(2 i-1)^{4}}$ is
(A) $\frac{14}{15} \lambda$
(B) $\frac{\lambda}{2}$
(C) $\frac{16}{15} \lambda$
(D) $\frac{15}{16} \lambda$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:51

Problem 14

The sum of all possible products of the first $n$ natural numbers taken two at a time is
(A) $\frac{1}{2}\left[\Sigma n^{2}-\Sigma n\right]$
(B) $\frac{1}{2}\left[(\Sigma n)^{2}-\Sigma n\right]$
(C) $\frac{1}{2}\left[\Sigma n^{2}-\Sigma(n+1)\right]$
(D) $\frac{1}{2}\left[(\Sigma n)^{2}-\Sigma n^{2}\right]$

Wendi Zhao
Wendi Zhao
Numerade Educator
03:59

Problem 15

The minimum value of $8^{\sin x^{\prime} 8}+8^{\cos x^{\prime} 8}$ is
(A) $2^{\frac{1}{3-\sqrt{2} / \sqrt{2}}}$ (B)
$2^{\frac{3+\sqrt{2}}{\sqrt{2}}}$
(C) $2^{\frac{1}{3+\sqrt{2} / \sqrt{2}}}$ (D)
$2^{\frac{3-\sqrt{2}}{\sqrt{2}}}$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:16

Problem 16

If $\log _{2^{12}} a+\log _{2^{n}} a+\log _{2^{n}} a+\log _{2^{n}} a+\ldots$ upto 20
terms is 840 , then $a$ is equal to(A) 2
(B) 1
(C) 4
(D) $\sqrt{2}$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:22

Problem 17

Sum to $n$ terms of the series
$\frac{1}{(1+x)(1+2 x)}+\frac{1}{(1+2 x)(1+3 x)}$ is
(A) $\frac{n x}{(1+x)(1+n x)}$
(B) $\frac{n}{(1+x)[1+(n+1) x]}$
(C) $\frac{x}{(1+x)(1+(n-1) x)}$
(D) None of these

mp
Manik Pulyani
Numerade Educator
02:10

Problem 18

If $a, b, c$ are distinct positive real numbers and $a^{2}+b^{2}$ $+c^{2}=1$, then $a b+b c+c a$ is
(A) less than 1
(B) equal to 1
(C) greater than 1
(D) any real number

Wendi Zhao
Wendi Zhao
Numerade Educator
02:01

Problem 19

The value of $(n-2)^{2}+(n-4)^{2}+(n-6)^{2}+\ldots$ to $n$ terms is
(A) $\frac{n}{3}\left(n^{2}+2\right)$
(B) $\frac{n}{2}\left(n^{2}+3\right)$
(C) $\frac{n}{3}\left(n^{2}-2\right)$
(D) $\frac{n}{2}\left(n^{2}-3\right)$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:55

Problem 20

The sum to infinity of the series $1+2\left(1-\frac{1}{n}\right)+3\left(1-\frac{1}{n}\right)^{2}+\ldots$ where $n \in N$, is given
by
(A) $n(n-1)$
(B) $n\left(1-\frac{1}{n}\right)^{2}$
(C) $n^{2}$
(D) $\left(\frac{n-1}{n}\right)^{2}$

Wendi Zhao
Wendi Zhao
Numerade Educator
07:51

Problem 21

$a_{1}, a_{2}, a_{3}, \ldots$ are in A.P. with common difference not a multiple of 3 . Then, maximum number of consecutive terms so that all the terms are prime numbers is
(A) 2
(B) 3
(C) 5
(D) infinite

Aflah M
Aflah M
Numerade Educator
01:27

Problem 22

The coefficient of $x^{49}$ in the product $(x-1)(x-3) \ldots$ $(x-99)$ is
(A) $-99^{2}$
(B) 1
(C) $-2500$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
01:44

Problem 23

If $x, y, z$ are three real numbers of the same sign then the value of $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$ lies in the interval
(A) $[2, \infty)$
(B) $[3, \infty)$
(C) $(3, \infty)$
(D) $(-\infty, 3)$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:43

Problem 24

In a G.P. of alternating positive and negative terms, any term is the A.M. of the next two terms. Then the common ratio is
(A) $-1$
(B) $-3$
(C) $-2$
(D) $\frac{-1}{2}$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:33

Problem 25

If $A=1+r^{a}+r^{2 a}+r^{3 a}+\ldots .$ as and $B=1+r^{b}+r^{2 b}+$
$r^{3 b}+\ldots .$ as, then $\frac{a}{b}$ is equal to
(A) $\log _{B}^{A}$
(B) $\log _{1-B}^{(1-A)}$
(C) $\log _{\frac{B-1}{B}}\left(\frac{A-1}{A}\right)$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
03:21

Problem 26

If the sum of $n$ terms of an A.P. is cn $(n-1)$, where $c \neq 0$, then sum of the squares of these terms is
(A) $c^{2} n^{2}(n+1)^{2}$
(B) $\frac{2}{3} c^{2} n(n-1)(2 n-1)$
(C) $\frac{2 c^{2}}{3} n(n+1)(2 n+1)$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
01:27

Problem 27

If in an A.P., $S_{n}=p, n^{2}$ and $S_{m}=p, m^{2}$ where $S_{p}$ denotes the sum of $r$ terms of the A.P., then $S_{p}$ is equal to
(A) $\frac{1}{2} p^{3}$
(B) $m m p$
(C) $p^{3}$
(D) $(m+n) p^{2}$

mp
Manik Pulyani
Numerade Educator
02:00

Problem 28

If $b_{1}, b_{2}$ and $b_{3}\left(b_{1}>0\right)$ are three successive terms of a
G.P. with common ratio $r$, the value of $r$ for which the inequality $b_{3}>4 b_{2}-3 b_{1}$ holds, is given by
(A) $r>3$
(B) $r<1$
(C) $r=2.5$
(D) $r=1.7$

Wendi Zhao
Wendi Zhao
Numerade Educator
03:22

Problem 29

If $p, q, r$ are positive and are in A.P., the roots of quadratic equation $p x^{2}+q x+r=0$ are all real for
(A) $\left|\frac{r}{p}-7\right| \geq 4 \sqrt{3}$
(B) $\left|\frac{p}{r}-7\right| \geq 4 \sqrt{3}$
(C) all $p$ and $r$
(D) no $p$ and $r$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:27

Problem 30

The sum to $n$ terms of the series $\frac{1}{3}+\frac{5}{9}+\frac{19}{27}+\frac{65}{81}+\ldots$ is
(A) $n-\frac{\left(3^{n}-2^{n}\right)}{2^{n}}$
(B) $n-\frac{2\left(3^{n}-2^{n}\right)}{3^{n}}$
(C) $2^{n}-1$
(D) $3^{n}-1$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:49

Problem 31

Sum to $n$ terms of the series $\frac{1}{5 !}+\frac{1 !}{6 !}+\frac{2 !}{7 !}+\frac{3 !}{8 !}+\ldots$ is
(A) $\frac{2}{5 !}-\frac{1}{(n+1) !}$
(B) $\frac{1}{4}\left(\frac{1}{4 !}-\frac{n !}{(n+4) !}\right)$
(C) $\frac{1}{4}\left(\frac{1}{3 !}-\frac{3 !}{(n+2) !}\right)$
(D) None of these

mp
Manik Pulyani
Numerade Educator
02:01

Problem 32

If $a, b, c, d$ and $p$ are distinct real numbers such that $\left(a^{2}+b^{2}+c^{2}\right) p^{2}-2 p(a b+b c+c d)+\left(b^{2}+c^{2}+d^{2}\right)$
$\leq 0$ then $a, b, c, d$ are in
(A) A.P.
(B) G.P.
(C) H.P.
(D) $a b=c d$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:50

Problem 33

If $a+b+c=3$ and $a>0, b>0, c>0$, then the greatest value of $a^{2} b^{3} c^{2}$ is
(A) $\frac{3^{10} \cdot 2^{4}}{7^{7}}$
(B) $\frac{3^{9} \cdot 2^{4}}{7^{7}}$
(C) $\frac{3^{8} \cdot 2^{4}}{7^{7}}$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
02:03

Problem 34

If $\left|\begin{array}{ccc}a & b & a \alpha-b \\ b & c & b \alpha-c \\ 2 & 1 & 0\end{array}\right|=0$ and $\alpha \neq \frac{1}{2}$, then
(A) $a, b, c$ are in A.P.
(B) $a, b, c$ are in G.P.
(C) $a, b, c$ are in H.P.
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
03:02

Problem 35

Suppose $a, b, c$ are in A.P. and $a^{2}, b^{2}, c^{2}$ are in G.P. If $a<b<c$ and $a+b+c=\frac{3}{2}$, then the value of $a$ is
(A) $\frac{1}{2 \sqrt{2}}$
(B) $\frac{1}{2 \sqrt{3}}$
(C) $\frac{1}{2}-\frac{1}{\sqrt{3}}$
(D) $\frac{1}{2}-\frac{1}{\sqrt{2}}$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:58

Problem 36

If $a_{1}, a_{2}, \ldots, a_{n}$ are in A.P. with common difference $d \neq 0$, then sum of the series $\sin d\left[\sec a_{1} \sec a_{2}+\sec \right.$
$\left.a_{2} \sec a_{3}+\ldots+\sec a_{n-1} \sec a_{n}\right]$ is
(A) $\tan a_{n}-\tan a_{1}$
(B) $\cot a_{n}-\cot a_{1}$
(C) $\sec a_{n}-\sec a_{1}$
(D) $\operatorname{cosec} a_{n}-\operatorname{cosec} a_{1}$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:12

Problem 37

The first and last term of an A.P. are $a$ and $l$ respectively. If $S$ is the sum of all the terms of the A.P. and the common difference is $\frac{l^{2}-a^{2}}{k-(l+a)}$, then $k$ is equal to
(A) $S$
(B) $2 S$
(C) $3 S$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
01:52

Problem 38

If $a, b, c, d$ are in G.P., then $\left(a^{2}+b^{2}+c^{2}\right)\left(b^{2}+c^{2}+d^{2}\right)=$
(A) $(a b+a c+b c)^{2}$
(B) $(a c+c d+a d)^{2}$
(C) $(a b+b c+c d)^{2}$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
01:06

Problem 39

If one geometric mean $G$ and two arithmetic means $p$ and $q$ be inserted between two numbers, then $G^{2}$ is equal to
(A) $(3 p-q)(3 q-p)$
(B) $(2 p-q)(2 q-p)$
(C) $(4 p-q)(4 q-p)$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
01:21

Problem 40

The product of $n$ positive integers is 1 , then their sum is a positive integer, that is
(A) equal to $]$
(B) equal to $n+n^{2}$
(C) divisible by $n$
(D) never less than $n$

Wendi Zhao
Wendi Zhao
Numerade Educator
03:18

Problem 41

A man saves ? 200 in each of the first three months of his service. In each of the subsequent months his saving increases by ? 40 more than the saving of immediately previous months. His total saving from the start of service will be ? 11040 after
(A) 21 months
(B) 18 months
(C) 19 months
(D) 20 months

Wendi Zhao
Wendi Zhao
Numerade Educator
03:21

Problem 42

Statement-1: The sum of the series $1+(1+2+4)+$ $(4+6+9)+(9+12+16)+\ldots+(361+380+400)$ is
$8000 .$
$\begin{array}{l}\text { Statement-2: } \\ \text { number } n .\end{array}_{k=1}^{n}\left(k^{3}-(k-1)^{3}\right)=n^{3}$, for any natural
(A) Statement- 1 is false, Statement-2 is true.
(B) Statement- 1 is true, statement- 2 is true; statement- 2 is a correct explanation for Statement- 1
(C) Statement- 1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1.
(D) Statement- 1 is true, statement- 2 is false.

Wendi Zhao
Wendi Zhao
Numerade Educator
01:23

Problem 43

If 100 times the $100^{\text {th }}$ term of an $A P$ with non-zero common difference equals the 50 times its $50^{\text {th }}$ term, then the $150^{\text {th }}$ term of this $A P$ is
(A) $-150$
(B) 150 times its $50^{\text {th }}$ term
(C) 150
(D) zero

Wendi Zhao
Wendi Zhao
Numerade Educator
03:19

Problem 44

If the sum of first $n$ terms of two A.P's are in the ratio $3 n+8: 7 n+15$, then the ratio of their 12 th terms is
(A) $8: 7$
(B) $7: 16$
(C) $74: 169$
(D) $13: 47$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:47

Problem 45

The sum of $n$ terms of the series $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\ldots$
(A) $2^{n}-n-\frac{1}{2}$
(B) $1-2^{-n}$
(C) $n+2^{-n}-1$
(D) $\frac{1}{2}\left(2^{n}-1\right)$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:56

Problem 46

The first two terms of a geometric progression add up to 12 . The sum of the third and the fourth terms is 48 . If the terms of the geometric progression are alternately positive and negative, then the first term is
(A) $-4$
(B) $-12$
(C) 12
(D) 4

Wendi Zhao
Wendi Zhao
Numerade Educator
02:38

Problem 47

The sum to the infinity of the series $1+\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{2}}+\frac{14}{3^{4}}+\ldots$ is
(A) 2
(B) 3
(C) 4
(D) 6

Wendi Zhao
Wendi Zhao
Numerade Educator
02:36

Problem 48

The sum of positive terms of the series
$10+9 \frac{4}{7}+9 \frac{1}{7}+\ldots$ is
(A) $\frac{352}{7}$
(B) $\frac{437}{7}$
(C) $\frac{852}{7}$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
00:51

Problem 49

The sum of the products of the $2 n$ numbers $\pm 1, \pm 2, \pm 3$. $\ldots . \pm n$ taking two at a time is
(A) $\frac{n(n+1)}{2}$
(B) $-\frac{n(n+1)}{2}$
(C) $\frac{n(n+1)(2 n+1)}{6}$
(D) $-\frac{n(n+1)(2 n+1)}{6}$

mp
Manik Pulyani
Numerade Educator
02:02

Problem 50

If $a$ is the first term, $d$ the common difference and $S_{k}$ the sum to $k$ terms of an A.P., then for $\frac{S_{k x}}{S_{x}}$ to be inde- pendent of $x$
(A) $a=2 d$
(B) $a=d$
(C) $2 a=d$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
03:11

Problem 51

Given that $\alpha, \gamma$ are roots of the equation $A x^{2}-4 x+1=0$ and $\beta, \delta$ are roots of the equation $B x^{2}-6 x+1=0$. If $\alpha, \beta, \gamma$ and $\delta$ are in H.P., then
(A) $A=5$
(B) $A=-3$
(C) $B=8$
(D) $B=-8$

Wendi Zhao
Wendi Zhao
Numerade Educator
03:05

Problem 52

The sum of $n$ terms of $m$ A.P.s are $S_{1}, S_{2}, S_{3}, \ldots, S_{m}$, If the first term and common difference are $1,2,3, \ldots, m$ respectively, then $S_{1}+S_{2}+S_{3}+\ldots+S_{m}=$
(A) $\frac{1}{4} m n(m+1)(n+1)$
(B) $\frac{1}{2} m n(m+1)(n+1)$
(C) $m n(m+1)(n+1)$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
02:37

Problem 53

If three positive numbers $a, b, c$ are in H.P., then $a^{n}+c^{n}$
$(\mathrm{A})>2 b^{n}$
$(\mathrm{B})=2 b^{n}$
$(\mathrm{C})<2 b^{n}$
$(\mathrm{D})>b^{n}$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:58

Problem 54

The sum of first $n$ terms of the series
$1 \cdot 1 !+2 \cdot 2 !+3 \cdot 3 !+4 \cdot 4 !+\ldots$ is
(A) $(n+1) !-1$
(B) $n !-1$
(C) $(n-1) !-1$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
01:40

Problem 55

If $a, b, c$ are digits, then the rational number represented by $0 \cdot c a b a b a b \ldots$ is
(A) $\frac{99 c+a b}{990}$
(B) $\frac{99 c+10 a+b}{99}$
(C) $\frac{99 c+10 a+b}{990}$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
02:17

Problem 56

The sum of first $n$ terms of the series
$1^{2}+2.2^{2}+3^{2}+2.4^{2}+5^{2}+5.6^{2}+\ldots$ is $\frac{n(n+1)^{2}}{2}$
when $n$ is even. When $n$ is odd, the sum is
(A) $\frac{n^{2}(n+1)}{2}$
(B) $\frac{n(n+1)^{2}}{2}$
(C) $\left[\frac{n(n+1)}{2}\right]^{2}$
(D) $\frac{n(n+1)}{2}$

Wendi Zhao
Wendi Zhao
Numerade Educator
03:12

Problem 57

The sum of the series $1+2 \cdot 2+3 \cdot 2^{2}+4 \cdot 2^{3}+5 \cdot 2^{4}+\ldots+100 \cdot 2^{99}$ is
(A) $99 \cdot 2^{100}+1$
(B) $100 \cdot 2^{100}$
(C) $99 \cdot 2^{100}$
(D) $99 \cdot 2^{100}+1$

Wendi Zhao
Wendi Zhao
Numerade Educator
09:06

Problem 58

Four different integers form an increasing A.P. If one of these numbers is equal to the sum of the squares of the other three numbers, then the numbers are
(A) $-2,-1,0,1$
(B) $0,1,2,3$
(C) $-1,0,1,2$
(D) None of these

Aflah M
Aflah M
Numerade Educator
03:52

Problem 59

If three successive terms of a G.P. with common ratio $r(r>1)$ form the sides of a $\Delta A B C$ and $[r]$ denotes greatest integer function, then $[r]+[-r]=$
(A) 0
(B) 1
(C) $-1$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
01:59

Problem 60

Let $S_{n}(1 \leq n \leq 9)$ denotes the sum of $n$ terms of series $1+22+333+\ldots+999999999$, then for $2 \leq n \leq 9$
(A) $S_{n}-S_{n-1}=\frac{1}{9}\left(10^{n}-n^{2}+n\right)$
(B) $S_{n}=\frac{1}{9}\left(10^{n}-n^{2}+2 n-2\right)$
(C) $9\left(S_{n}-S_{n-1}\right)=n\left(10^{n}-1\right)$
(D) None of these

mp
Manik Pulyani
Numerade Educator
03:12

Problem 61

$a, b, c$ are three distinct real numbers, which are in G.P. and $a+b+c=x b$. Then,
(A) $x<-1$ or $x>3$
(B) $-1<x<3$
(C) $-1<x<2$
(D) $0<x<1$

Wendi Zhao
Wendi Zhao
Numerade Educator
03:46

Problem 62

If $a_{1}, a_{2}, a_{3}, a_{4}$ are in H.P., then $\frac{1}{a_{1} a_{4}} \sum_{r=1}^{3} a_{r} a_{r+1}$ is a
root of
(A) $x^{2}+2 x+15=0$
(B) $x^{2}+2 x-15=0$
(C) $x^{2}-6 x-8=0$
(D) $x^{2}-9 x+20=0$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:15

Problem 63

The sum to $n$ terms of the series $\frac{1}{3}+\frac{5}{9}+\frac{19}{27}+\frac{65}{81}+\ldots$ is
(A) $n-\frac{\left(3^{n}-2^{n}\right)}{2^{n}}$
(B) $n-\frac{2\left(3^{n}-2^{n}\right)}{3^{n}}$
(C) $2^{n}-1$
(D) $3^{n}-1$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:37

Problem 64

If $a+b+c=3$ and $a>0, b>0, c>0$, then the greatest value of $a^{2} b^{3} c^{2}$ is
(A) $\frac{3^{10} \cdot 2^{4}}{7^{7}}$
(B) $\frac{3^{9} \cdot 2^{4}}{7^{7}}$
(C) $\frac{3^{8} \cdot 2^{4}}{7^{7}}$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
02:24

Problem 65

Let the harmonic mean and the geometric mean of two positive numbers be in the ratio $4: 5$. The two numbers are in the ratio
(A) $1: 1$
(B) $2: 1$
(C) $3: 1$
(D) $4: 1$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:45

Problem 66

The first and last term of an A.P. are $a$ and $l$, respectively. If $S$ is the sum of all the terms of the A.P. and the common difference is $\frac{l^{2}-a^{2}}{k-(l+a)}$, then $k$ is equal to
(A) $S$
(B) $2 S$
(C) $3 S$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
02:58

Problem 67

If $a_{1}, a_{2}, \ldots, a_{n}$ are in A.P. with common difference $d \neq 0$, then sum of the series $\sin d\left[\sec a_{1} \sec a_{2}+\sec a_{2}\right.$
$\left.\sec a_{3}+\ldots+\sec a_{n-1} \sec a_{n}\right]$ is
(A) $\tan a_{n}-\tan a_{1}$
(B) $\cot a_{n}-\cot a_{1}$
(C) $\sec a_{n}-\sec a_{1}$
(D) $\operatorname{cosec} a_{n}-\operatorname{cosec} a_{1}$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:21

Problem 68

Sum to $n$ terms of the series $\frac{1}{5 !}+\frac{1 !}{6 !}+\frac{2 !}{7 !}+\frac{3 !}{8 !}+\ldots$ is
(A) $\frac{2}{5 !}-\frac{1}{(n+1) !}$
(B) $\frac{1}{4}\left(\frac{1}{4 !}-\frac{n !}{(n+4) !}\right)$
(C) $\frac{1}{4}\left(\frac{1}{3 !}-\frac{3 !}{(n+2) !}\right)$
(D) None of these

mp
Manik Pulyani
Numerade Educator
03:10

Problem 69

If $\left\langle a_{n}\right\rangle$ and $\left\langle b_{n}\right\rangle$ be two sequences given by $a_{n}=(x)^{1 / 2^{*}}+(y)^{1 / 2^{*}}$ and $b_{n}=(x)^{1 / 2^{*}}-(y)^{1 / 2^{*}}$ for all
$n \in N$. Then, $a_{1} a_{2} a_{3} \ldots a_{n}$ is equal to
(A) $x-y$
(B) $\frac{x+y}{b_{n}}$
(C) $\frac{x-y}{b_{n}}$
(D) $\frac{x y}{b_{n}}$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:40

Problem 70

For any odd integer $n \geq 1$, $n^{3}-(n-1)^{3}+\ldots+(-1)^{n-1} 1^{3}=$
(A) $\frac{1}{2}(n-1)^{2}(2 n-1)$
(B) $\frac{1}{4}(n-1)^{2}(2 n-1)$
(C) $\frac{1}{2}(n+1)^{2}(2 n-1)$
(D) $\frac{1}{4}(n+1)^{2}(2 n-1)$

mp
Manik Pulyani
Numerade Educator
01:48

Problem 71

For a positive integer $n$, let $a(n)=$ $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots+\frac{1}{\left(2^{n}\right)-1}$, Then
(A) $a(100) \leq 100$
(B) $a(100)>100$
(C) $a(200) \leq 100$
(D) $a(200)>100$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:41

Problem 72

Let $\alpha, \beta, \gamma$ be the roots of the equation $3 x^{3}-x^{2}-3 x+1=0 .$ If $\alpha, \beta, \gamma$ are in H.P. then $|\alpha-\gamma|=$
(A) $\frac{1}{3}$
(B) $\frac{2}{3}$
(C) $\frac{4}{3}$
(D) None of these

mp
Manik Pulyani
Numerade Educator
01:13

Problem 73

Suppose $a, b>0$ and $x_{1}, x_{2}, x_{3}\left(x_{1}>x_{2}>x_{3}\right)$ are roots of $\frac{x-a}{b}+\frac{x-b}{a}=\frac{b}{x-a}+\frac{a}{x-b}$ and $x_{1}-x_{2}-x_{3}=c$,
then $a, b, c$ are in
(A) A.P
(B) $\mathrm{G} . \mathrm{P}_{4}$
(C) H.P.
(D) None of these

mp
Manik Pulyani
Numerade Educator
01:19

Problem 74

The coefficient of $x^{n}$ in the product $(1-x)(1-2 x)\left(1-2^{2} \cdot x\right)\left(1-2^{3} \cdot x\right) \ldots\left(1-2^{n}+x\right)$ is
equal to
$($ A $)\left(1-2^{n+1}\right) 2^{\frac{n(n-1)}{2}}$.
(B) $\left(2^{n+1}-1\right) \cdot 2^{\frac{n(n-1)}{2}}$
(C) $\left(1-2^{n}\right) 2^{\frac{n(n-1)}{2}}$.
(D) None of these

mp
Manik Pulyani
Numerade Educator
01:39

Problem 75

If $0.272727 \ldots, x$ and $0.727272 \ldots$ are in H.P., then $x$ must be
(A) rational
(B) integer
(C) irrational
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
03:32

Problem 76

If $a_{1}=0$ and $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ are real numbers such that $\left|a_{i}\right|=\left|a_{i-1}+1\right|$ for all $i$ then the A.M. of the numbers $a_{1}, a_{2}, \ldots, a_{n}$ has value $x$ where
(A) $x \leq-\frac{1}{2}$
(B) $x \geq-\frac{1}{2}$
(C) $x<-\frac{1}{2}$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
03:44

Problem 77

If $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ are in H.P, then
$\frac{a_{1}}{a_{2}+a_{3}+\ldots+a_{n}}, \frac{a_{2}}{a_{1}+a_{3}+\ldots+a_{n}} \cdots$
$\frac{a_{n}}{a_{1}+a_{2}+\ldots+a_{n-1}}$ are in
(A) A.P.
(B) G.P.
(C) H.P.
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
05:04

Problem 78

The consecutive numbers of a three digit number form a G.P. If we subtract 792 from this number, we get a number consisting of the same digits written in the reverse order and if we increase the second digit of the required number by 2, the resulting number forms an
A.P. The number is
(A) 139
(B) 193
(C) 931
(D) None of these

Aflah M
Aflah M
Numerade Educator
02:03

Problem 79

The largest term of the sequence $\frac{1}{503}, \frac{4}{524}, \frac{9}{581}, \frac{16}{692}, \ldots$ is
(A) $\frac{16}{692}$
(B) $\frac{4}{524}$
(C) $\frac{49}{1520}$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
01:46

Problem 80

The coefficient of $x^{99}$ and $x^{98}$ in the polynomial $(x-1)(x-2)(x-3) \ldots(x-100)$ are
(A) $-5050$ and 12482075
(B) $-4050$ and 12582075
(C) $-5050$ and 12582075
(D) None of these

mp
Manik Pulyani
Numerade Educator
01:17

Problem 81

The three successive terms of a G.P. will form the sides of a triangle if the common ratio $r$ satisfies the inequality
(A) $\frac{\sqrt{3}-1}{2}<r<\frac{\sqrt{3}+1}{2}$
(B) $\frac{\sqrt{5}-1}{2}<r<\frac{\sqrt{5}+1}{2}$
(C) $\frac{\sqrt{2}-1}{2}<r<\frac{\sqrt{2}+1}{2}$
(D) None of these

mp
Manik Pulyani
Numerade Educator
10:59

Problem 82

If the sides of a right angled triangle are in G.P., then the cosine of the greater acute angle is
(A) $\frac{1}{1+\sqrt{5}}$
(B) $\frac{1}{1-\sqrt{5}}$
(C) $\frac{1+\sqrt{5}}{2}$
(D) None of these

Aflah M
Aflah M
Numerade Educator
01:13

Problem 83

Sum to $n$ terms of the series $2+5+14+41+\ldots$ is
(A) $\frac{n}{2}+\frac{1}{4}\left(3^{n}-1\right)$
(B) $\frac{n}{2}+\frac{3}{4}\left(3^{n}-1\right)$
(C) $\frac{n}{2}+\frac{1}{2}\left(3^{n}-1\right)$
(D) None of these

mp
Manik Pulyani
Numerade Educator
01:44

Problem 84

If the $p t h, q$ th and $r$ th terms of both an A.P. and a G.P. be respectively $a, b$ and $c$, then
(A) $a^{e} \cdot c^{b} \cdot b^{a}=a^{c} \cdot b^{e} \cdot a^{b}$
(B) $a^{b-1} \cdot b^{e+1} \cdot c^{a-1}=a^{e-1} \cdot b^{a-1} \cdot c^{b+1}$
(C) $a^{b} \cdot b^{c} \cdot c^{a}=a^{c} \cdot b^{a} \cdot c^{b}$
(D) None of these

mp
Manik Pulyani
Numerade Educator
08:31

Problem 85

If, in a G.P. of $3 n$ terms, $S_{1}$ denotes the sum of the first $n$ terms, $S_{2}$ the sum of the second block of $n$ terms and $S_{3}$ the sum of the last $n$ terms, then $S_{1}, S_{2}, S_{3}$ are in
(A) A.P.
(B) G.P.
(C) H.P.
(D) None of these

Aflah M
Aflah M
Numerade Educator
00:56

Problem 86

In a geometric series, the first term is $a$ and common ratio is $r$. If $\mathrm{S}_{n}$ denotes the sum of $n$ terms and $U_{n}$ $=\sum_{n=1}^{n} \mathrm{~S}_{n}$, then $r S_{n}+(1-r) u_{n}=$
(A) $n a$
(B) $(n-1) a$
(C) $(n+1) a$
(D) None of these

mp
Manik Pulyani
Numerade Educator
08:21

Problem 87

In a $\Delta a b c$, if $\cot A, \cot B, \cot C$ are in A.P. then $a^{2}, b^{2}$,
$c^{2}$ are in
(A) A.P.
(B) G.P.
(C) H.P.
(D) A,G. P,

Aflah M
Aflah M
Numerade Educator
02:03

Problem 88

If $\frac{1}{1^{4}}+\frac{1}{2^{4}}+\frac{1}{3^{4}}+\ldots .$ up to $\infty=\frac{\pi^{4}}{90}$, then the value of
$\frac{1}{1^{4}}+\frac{1}{3^{4}}+\frac{1}{5^{4}}+\ldots .$ up to $\infty$ is
(A) $\frac{\pi^{4}}{45}$
(B) $\frac{\pi^{4}}{96}$
(C) $\frac{\pi^{4}}{124}$
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
01:40

Problem 89

If the $(m+1)$ th, $(n+1)$ th and $(r+1)$ th terms of an A.P. are in G.P. and $m, n, r$ are in H.P., then the ratio of the first term of the A.P. to its common difference is
(A) $\frac{n}{3}$
(B) $-\frac{n}{3}$
(C) $\frac{n}{2}$
(D) $-\frac{n}{2}$

mp
Manik Pulyani
Numerade Educator
01:43

Problem 90

Let there be $n$ numbers in G.P. whose common ratio is $r$ and $S_{m}$ denotes the sum of their first $m$ terms. The sum of their products taken two at a time is $k S_{n} S_{n-1}$ where $k=$
(A) $\frac{r-1}{r}$
(B) $\frac{r-1}{r+1}$
(C) $\frac{r}{r+1}$
(D) None of these

mp
Manik Pulyani
Numerade Educator
02:44

Problem 91

If $a, b, c_{3} d$ are distinct integers in A.P. such that $d=a^{2}$ $+b^{2}+c^{2}$, then $a+b+c+d=$
(A) 2
(B) 1
(C) 0
(D) None of these

mp
Manik Pulyani
Numerade Educator
01:03

Problem 92

If $H_{n}=1+\frac{1}{2}+\frac{1}{3}+\ldots .+\frac{1}{n}$, then the value of
$1+\frac{3}{2}+\frac{5}{3}+\ldots+\frac{2 n-1}{n}$ is
(A) $n-H_{n}$
(B) $2 n-H_{n}$
(C) $(n-1)-H_{n}$
(D) $n-2 H_{n}$

mp
Manik Pulyani
Numerade Educator
02:12

Problem 93

If $a_{m}$ be the $m$ th term of an A.P., then $a_{1}^{2}-a_{2}^{2}+a_{3}^{2}-a_{4}^{2}+\ldots .+a_{2 n-1}^{2}-a_{2 n}^{2}=$
(A) $\frac{n-1}{2 n-1}\left(a_{1}^{2}-a_{2 n}^{2}\right)$
(B) $\frac{n}{2 n-1}\left(a_{2 n}^{2}-a_{1}^{2}\right)$
(C) $\frac{n}{2 n-1}\left(a_{1}^{2}-a_{2 n}^{2}\right)$
(D) None of these

mp
Manik Pulyani
Numerade Educator
02:04

Problem 94

If $a_{n+1}=\frac{1}{1-a_{n}}$ for $n \geq 1$ and $a_{3}=a_{1}$, then $\left(a_{2001}\right)^{2001}=$
(A) 1
(B) $-1$
(C) 0
(D) None of these

mp
Manik Pulyani
Numerade Educator
01:47

Problem 95

If $a, b, c$ are positive numbers in G.P. and log $\left(\frac{5 c}{a}\right), \log \left(\frac{3 b}{5 c}\right)$ and $\log \left(\frac{a}{3 b}\right)$ are in A.P. then $a, b, c$
(A) form the sides of an equilateral triangle
(B) form the sides of an isosceles triangle
(C) form the sides of a right angled triangle
(D) can not form the sides of a triangle

mp
Manik Pulyani
Numerade Educator
01:56

Problem 96

If $a, b, c$ are in G.P. and $\log a-\log 2 b, \log 2 b-\log 3 c$
and $\log 3 c-\log a$ are in A.P., then $a, b, c$ are the sides of a triangle which is
(A) right angled
(B) acute angled
(C) obtuse angled
(D) None of these

mp
Manik Pulyani
Numerade Educator
01:42

Problem 97

In a sequence of $4 n+1$ terms, the first $2 n+1$ terms are in A.P. having common difference 2 and the last $2 n+1$ terms are in G.P. having common ratio $\frac{1}{2}$, If the middle term of the A.P. is equal to the middle term of the G.P. then the middle term of the sequence is
(A) $\frac{n \cdot 2^{n+1}}{2^{n}+1}$
(B) $\frac{n \cdot 2^{n+1}}{2^{n}-1}$
(C) $\frac{n \cdot 2^{n}}{2^{n}-1}$
(D) None of these

mp
Manik Pulyani
Numerade Educator
02:10

Problem 98

If $S_{1}, S_{2}$ and $S_{3}$ denote the sums up to $n>1$ terms of three sequences in A.P. whose first terms are unity and common differences are in H.P. then $n=$
(A) $\frac{2 S_{3} S_{1}+S_{1} S_{2}+S_{2} S_{3}}{S_{1}-2 S_{2}+S_{3}}$
(B) $\frac{2 S_{3} S_{1}-S_{1} S_{2}-S_{2} S_{3}}{S_{1}+2 S_{2}+S_{3}}$
(C) $\frac{2 S_{3} S_{1}-S_{1} S_{2}-S_{2} S_{3}}{S_{1}-2 S_{2}+S_{3}}$
(D) None of these

mp
Manik Pulyani
Numerade Educator
02:12

Problem 99

Sum to $n$ terms of the series $1^{3}+3.2^{3}+3^{3}+3.4^{3}+5^{3}$ $+\ldots .$, where $n$ is even, is
(A) $\frac{n^{2}\left(n^{2}-3 n+1\right)}{2}$
(B) $\frac{n^{2}\left(n^{2}+3 n+1\right)}{2}$
(C) $\frac{n^{2}\left(n^{2}+3 n+1\right)}{4}$
(D) None of these

mp
Manik Pulyani
Numerade Educator
01:37

Problem 100

Let $a$ be a fixed real number such that
$\frac{a-x}{p x}=\frac{a-y}{q y}=\frac{a-z}{r z}$If $p, q, \mathrm{r}$ are in A.P. then $x, y, z$ are in
(A) A.P.
(B) G.P.
(C) H. P
(D) None of these

mp
Manik Pulyani
Numerade Educator
01:21

Problem 101

If $|a|<1$ and $|b|<1$, then the sum of the series $1+(1+a) b+\left(1+a+a^{2}\right) b^{2}+\left(1+a+a^{2}+a^{3}\right) b^{3}+$
$\ldots \infty$ is equal to
(A) $\frac{1}{(1-b)(1-a b)}$
(B) $\frac{1}{(1-a)(1-a b)}$
(C) $\frac{1}{(1-a)(1-b)}$
(D) None of these

mp
Manik Pulyani
Numerade Educator
01:39

Problem 102

If $<a_{n}>$ and $\left\langle b_{n}>\right.$ be two sequences given by $a_{n}=$ $x^{2-}+y^{2}$ and $b_{n}=x^{2^{-4}}-y^{2^{-4}} \forall n \in N$, then the
value of $a_{1} a_{2} a_{3} \ldots a_{n}$ is
(A) $\frac{x+y}{b_{n}}$
(B) $\frac{x-y}{b_{n}}$
(C) $\frac{x^{2}+y^{2}}{b_{n}}$
(D) $\frac{x^{2}-y^{2}}{b_{n}}$

mp
Manik Pulyani
Numerade Educator
View

Problem 103

The sixth term of an A.P. is equal to 2 . The value of the common difference of the A.P. which makes the product $a_{1} a_{4} a_{5}$ greatest, is
(A) $\frac{8}{5}$
(B) $\frac{2}{3}$
(C) $\frac{3}{5}$
(D) $\frac{3}{4}$

mp
Manik Pulyani
Numerade Educator
01:59

Problem 104

If the natural numbers are written as
$2^{1}$
456
$\begin{array}{llll}7 & 8 & 9 & 10\end{array}$
Then, the sum of the terms of the $n$th row is
(A) $\frac{n\left(n^{2}-1\right)}{2}$
(B) $\frac{n\left(n^{2}+1\right)}{4}$
(C) $\frac{n\left(n^{2}+1\right)}{2}$
(D) None of these

mp
Manik Pulyani
Numerade Educator
09:49

Problem 105

The H.M. of two numbers is 4 . If their A.M. $A$ and
G.M. $G$ satisfy the relation $2 A+G^{2}=27$, then the numbers are
(A) 1
(B) 2
(C) 3
(D) 6

Aflah M
Aflah M
Numerade Educator
01:19

Problem 106

If the first and the $(2 n-1)$ th terms of an A.P., G.P. and
H.P. are equal and their $n$th terms are $a, b, c$ respectively, then
(A) $a=b=c$
(B) $a \geq b \geq c$
(C) $a+c=b$
(D) $a c-b^{2}=0$

mp
Manik Pulyani
Numerade Educator
02:53

Problem 107

The real numbers $x_{1}, x_{2}, x_{3}$ satisfying the equation $x^{3}-x^{2}+b x+\gamma=0$ are in A.P. The intervals in which $\beta$ and $\gamma$ lie are
(A) $\beta \in\left(-\infty, \frac{1}{3}\right]$
(B) $\beta \in\left[-\frac{1}{27}, \infty\right)$
(C) $\gamma \in\left(-\infty, \frac{1}{3}\right]$
(D) $\gamma \in\left[-\frac{1}{27}, \infty\right)$

mp
Manik Pulyani
Numerade Educator
02:25

Problem 108

If $a, b, c$ are in A.P. and $a^{2}, b^{2}, c^{2}$ arc in H.P. then
(A) $a=b=c$
(B) $-\frac{a}{2}, b, c$ are in G.P.
(C) $-\frac{c}{2}, b, a$ are in G.P.
(D) $-\frac{a}{2}, b, c$ are in H.P.

mp
Manik Pulyani
Numerade Educator
02:12

Problem 109

If the G.M. between $a$ and $b$ be twice the H.M., then $\frac{a}{b}$ is equal to
(A) $\frac{2+\sqrt{3}}{2-\sqrt{3}}$
(B) $\frac{2-\sqrt{3}}{2+\sqrt{3}}$
(C) $\frac{4+\sqrt{3}}{4-\sqrt{3}}$
(D) $\frac{4-\sqrt{3}}{4+\sqrt{3}}$

mp
Manik Pulyani
Numerade Educator
02:30

Problem 110

If $a, b, c$ are in G.P. and $x$ is the A.M. between $a$ and $b, y$ the A.M. between $b$ and $c$, then
(A) $\frac{a}{x}+\frac{c}{y}=1$
(B) $\frac{a}{x}+\frac{c}{y}=2$
(C) $\frac{1}{x}+\frac{1}{y}=\frac{2}{b}$
(D) None of these

mp
Manik Pulyani
Numerade Educator
02:01

Problem 111

The solution of the equations $\log x+\log x^{1 / 2}+\log x^{1 / 4}$
$+\ldots=y$ and $\frac{1+3+5+\ldots . .+(2 y-1)}{4+7+10+\ldots+(3 y+1)}$
$=\frac{20}{7 \log x}$ is
(A) $x=10^{5}, 10^{-5 / 7}$
(B) $y=10,-\frac{10}{7}$
(C) $x=10,-\frac{10}{7}$
(D) $y=10^{5}, 10^{-5 / 7}$

mp
Manik Pulyani
Numerade Educator
01:30

Problem 112

The sum of of first ten terms of an A.P. is equal to 155 and the sum of first two terms of a G.P. is 9 . If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to the common difference of the A.P, then
(A) first term of the G.P. is $\frac{2}{3}, 3$
(B) first term of the A.P. is $\frac{2}{3}, 3$
(C) Common ratio of the G.P. is $\frac{25}{2}, 2$
(D) Common difference of the A.P is $\frac{2}{3}, 3$

mp
Manik Pulyani
Numerade Educator
01:49

Problem 113

Let $\left(1+x^{2}\right)^{2}(1+x)^{n}=\sum_{k=0}^{n+4} a_{k} x^{k}$. If $a_{1}, a_{2}, a_{3}$, are in
A.P., then $n$ is equal to
(A) 1
(B) 2
(C) 3
(D) 4

mp
Manik Pulyani
Numerade Educator
01:23

Problem 114

If $a, b, c$ are non-zero real numbers such that 3 $\left(a^{2}+b^{2}+c^{2}+1\right)=2(a+b+c+a b+b c+c a)$, then,
$a, b, c$ are in
(A) A.P.
(B) G. P.
(C) H.P.
(D) all equal

mp
Manik Pulyani
Numerade Educator
01:49

Problem 115

Let $t_{n}=\underbrace{1.1 \ldots 1}_{n \text { times }}$, then
(A) $t_{912}$ is not prime
(B) $t_{951}$ is not prime
(C) $t_{480}$ is not prime
(D) $t_{91}$ is not prime

mp
Manik Pulyani
Numerade Educator
01:22

Problem 116

Sum to $n$ terms of the series
$1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots$, is
(A) $\frac{n}{n+1}$
(B) $\frac{2 n}{n+1}$
(C) $\frac{n}{n-1}$
(D) None of these

mp
Manik Pulyani
Numerade Educator
00:47

Problem 117

Sum to infinite terms of the series
$1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots$ is
(A) 1
(B) 2
(C) 4
(D) None of these

mp
Manik Pulyani
Numerade Educator
01:51

Problem 118

Sum to $n$ terms of the series $\frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\ldots .$ is
(A) $\frac{n}{2 n+1}$
(B) $\frac{n}{2 n-1}$
(C) $\frac{n-1}{2 n+1}$
(D) None of these

mp
Manik Pulyani
Numerade Educator
00:50

Problem 119

Sum to infinite terms of the series $\frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\ldots .$ is
(A) $\frac{1}{4}$
(B) $\frac{1}{3}$
(C) $\frac{1}{2}$
(D) None of these

mp
Manik Pulyani
Numerade Educator
02:03

Problem 120

The sum to infinity of the series
$1+\frac{3}{2}+\frac{5}{2^{2}}+\frac{7}{2^{3}}+\ldots$ is
(A) 4
(B) 6
(C) 8
(D) None of these

mp
Manik Pulyani
Numerade Educator
01:30

Problem 121

If the sum to infinity of the series $3+5 r+7 r^{2}+\ldots$ is $\frac{49}{9}$, then $r$ is equal to
(A) $\frac{1}{4}$
(B) $\frac{1}{3}$
(C) $\frac{1}{2}$
(D) None of these

mp
Manik Pulyani
Numerade Educator
04:20

Problem 122

If the sum to infinity of the series $3+(3+d) \frac{1}{4}+(3+2 d) \frac{1}{4^{2}}+\ldots$ is $\frac{44}{9}$, then $d=$
(A) 1
(B) 2
(C) 4
(D) None of these

Ankit Singh
Ankit Singh
Numerade Educator
01:00

Problem 123

$3^{1 / 3} \cdot 9^{1 / 9} \cdot 27^{1 / 27} \cdot 81^{1 / 81} \ldots$ upto $\infty=$
(A) $\sqrt{27}$
(B) $\sqrt[3]{27}$
(C) $\sqrt[4]{27}$
(D) None of these

mp
Manik Pulyani
Numerade Educator
01:07

Problem 124

The sequence $\left\{S_{2 n}\right\}$ is
(A) increasing
(B) decreasing
(C) non-monotonic
(D) unbounded

mp
Manik Pulyani
Numerade Educator
01:07

Problem 125

The sequence $\left\{S_{2 n+1}\right\}$ is
(A) increasing
(B) decreasing
(C) non-monotonic
(D) unbounded

mp
Manik Pulyani
Numerade Educator
04:55

Problem 126

$S_{2 n+1}-S_{2 n}$ must be equal to
(A) $\left(\frac{1}{2}\right)^{2 n-1}$
(B) $\left(\frac{1}{2}\right)^{2 n-1}\left(S_{1}-S_{2}\right)$
(C) $\left(\frac{1}{2}\right)^{2 n-1}\left(S_{2}-S_{1}\right)$
(D) Zero

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:42

Problem 127

If $S_{1}>S_{2}$, then $\lim _{n \rightarrow \infty} S_{n}$ must be equal to
(A) $S_{1}-S_{2}$
(B) $S_{1}+2 S_{2}$
(C) $\frac{S_{1}+2 S_{2}}{3}$
(D) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:13

Problem 128

I. Let $S_{n}$ denotes the sum of $n$ terms of an A.P. whose first term is $a$.
(A) 29 If the common difference $d=S_{n}-k S_{n-1}+S_{n-2}$, then $k=$
II. The minimum number of terms from the beginning of the series
(B) 4 $20+22 \frac{2}{3}+25 \frac{1}{3}+\ldots$, so that the sum may exceed 1568 , is
III. If $5^{1+x}+5^{1-x}, \frac{a}{2}$ and $25^{x}+25^{-x}$ are three consecutive terms of an
(C) 2
A.P., then $a \geq k$, where $k=$
IV. If $\log _{2^{n}} a+\log _{2^{\text {? }}} a+\log _{2^{i n}} a+\log _{2^{n}} a+\ldots$ upto 20 terms is 840 ,
(D) 12 then $a$ is equal to...

Narayan Hari
Narayan Hari
Numerade Educator
05:19

Problem 129

Column-I Column
I. If the first term of an infinite G.P. is 1 and each term is twice the sum
(A) $\frac{2}{9}$
of the suceeding terms, then the common ratio is
II. Sum to infinity of the series $\frac{2}{3}-\frac{5}{6}+\frac{2}{3}-\frac{11}{24}+\ldots$ is
(B) $\frac{3}{2}$
III. $\lim _{n \rightarrow \infty}\left(1+3^{-1}\right)\left(1+3^{-2}\right)\left(1+3^{-4}\right)\left(1+3^{-8}\right) \ldots\left(1+3^{-2^{\prime}}\right)=$
(C) 1
IV. If $\sum_{k=1}^{n}\left(\sum_{m=1}^{k} m^{2}\right)=a n^{4}+b n^{3}+c n^{2}+d n+e$, then $a+b+c+d+e=$
(D) $\frac{1}{3}$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
08:30

Problem 130

\begin{tabular}{l}
Column-I & Column-II \\
\hline I. If $a, b, c$ are in A.P., $b, c, d$ are in G.P. and $c, d, e$ are in H.P., then (A) A.P. \\
$\qquad a, c, e$ are in \\
II. If $2(y-a)$ is the H.M. between $y-x, y-z$ then $x-a, y-a, z-a$ & (B) G.P. \\
are in \\
III. If three numbers are in H.P., then the numbers obtained by subtract- (C) H.P. \\
ing half of the middle number from each of them are in \\
IV. If $a, b, c$ are in G.P., then the equations $a x^{2}+2 b x+c=0$ and $d x^{2}+$ (D) A.G.P. \\
$2 e x+f=0$ have a common root, if $\frac{d}{a}, \frac{e}{b}$ and $\frac{f}{c}$ are in
\end{tabular}

Sandip Ranjan
Sandip Ranjan
Numerade Educator
09:49

Problem 131

Assertion: Between two numbers whose sum is, $2 \frac{1}{6}$ an even number of arithmetic means are inserted. If the sum of these means exceeds their number by unity, then the number of means are 12 Reason: If $a$ and $b$ are two given numbers and $A_{1}, A_{2}$, $\ldots, A_{n}$ are $n$ arithmetic means between them, then $A_{1}+A_{2}, \ldots, A_{n}=n\left(\frac{a+b}{2}\right)$

Sandip Ranjan
Sandip Ranjan
Numerade Educator
02:14

Problem 132

Assertion: If $a, b, c$ are distinct positive real numbers and $a^{2}+b^{2}+c^{2}=1$, then $a b+b c+c a$ is less than 1 . Reason: A.M. >G.M. for unequal numbers

Ankit Singh
Ankit Singh
Numerade Educator
02:14

Problem 133

Assertion: If $a, b, c, d \in R+$ and $a, b, c, d$ are in H.P., then $b+c>a+d$ Reason: H.M > A.M. for unequal numbers

Ankit Singh
Ankit Singh
Numerade Educator
09:01

Problem 134

Assertion: The sum of the series $\frac{1}{1+1^{2}+1^{4}}+\frac{2}{1+2^{2}+2^{4}}+\frac{3}{1+3^{2}+3^{4}}+\ldots$
to $n$ terms is $\frac{n(n+1)}{2\left(n^{2}+n+1\right)}$ Reason: The $n$th term of the above series is $T_{n}=\frac{1}{2}\left[\frac{1}{1+(n-1) n}-\frac{1}{1+n(n+1)}\right]$

Sandip Ranjan
Sandip Ranjan
Numerade Educator
07:41

Problem 135

Assertion: The value of $x+y+z$ is 15 if $a, x, y, z$ $b$ are in A.P., while the value of $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ is $\frac{5}{3}$ if $a, x, y, z, b$ are in H.P. The values of $a$ and $b$ are 9,1 respectively. Reason: The sum of $n$ A.M.s between two quantities is equal to $n$ times their single mean.

Sandip Ranjan
Sandip Ranjan
Numerade Educator
03:37

Problem 136

Assertion: For every natural number
$n,(n !)^{3}<n^{n}\left(\frac{n+1}{2}\right)^{2 n}$
Reason: A.M $>$ G.M. for $n$ distinct positive quantities

Bobby Barnes
Bobby Barnes
University of North Texas
03:09

Problem 137

If $1, \log _{3} \sqrt{\left(3^{1-x}+2\right)}, \log _{3}(4 \cdot 3 x-1)$ are in AP, then
$x$ equals:
(A) $\log _{3} 4$
(B) $1-\log _{3} 4$
(C) $1-\log , 3$
(D) $\log _{4} 3$

Aayush Gupta
Aayush Gupta
Numerade Educator
02:46

Problem 138

The value of $2^{1 / 4} \cdot 4^{1 / 8} \cdot 8^{1 / 16} \ldots \infty$ is:
(A) 1
(B) 2
(C) $3 / 2$
(D) 4

Wendi Zhao
Wendi Zhao
Numerade Educator
00:55

Problem 139

Fifth term of a GP is 2, then the product of its 9 terms is : $\quad[2002]$
(A) 256
(B) 512
(C) 1024
(D) None of these

Wendi Zhao
Wendi Zhao
Numerade Educator
02:11

Problem 140

Let $T_{n}$ denote the number of triangles which can be formed using the vertices of a regular polygon of $n$ sides. If $T_{n+1},=T_{n}=21$, then $n$ equals :
$\mid 20021$
(A) 5
(B) 7
(C) 6
(D) 4

Wendi Zhao
Wendi Zhao
Numerade Educator
04:06

Problem 141

The sum of the series $\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{3.4}-\ldots \ldots \ldots$ upto
1s equal to
(A) $2 \log _{e} 2$
(B) $\log _{2} 2-1$
(C) $\log _{e} 2$
(D) $\log _{e}\left(\frac{4}{e}\right)$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:44

Problem 142

If $f: R \rightarrow R$ satisfies $f(x+y)=f(x)+f(y)$, for all $x, y$ $\in R$ and $f(1)=7$, then $\sum_{r=1}^{n} f(r)$ is $|2003|$
(A) $\frac{7 n}{2}$
(B) $\frac{7(n+1)}{2}$
(C) $7 n(n+1)$
(D) $\frac{7 n(n+1)}{2}$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:29

Problem 143

If $S_{n}=\sum_{r=0}^{n} \frac{1}{{ }^{n} C_{r}}$ and $t_{n}=\sum_{r=0}^{n} \frac{r}{n_{r}}$, then $\frac{t_{n}}{S_{n}}$ is equal to
(A) $\frac{1}{2} n$
(B) $\frac{1}{2} n-1$
(C) $n-1$
(D) $\frac{2 n-1}{2}$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:05

Problem 144

Let $T_{r}$ be the $r$ th term of an A.P. whose first term is $\underline{a}$ and common difference is $d .$ If for some positive integers $m, n, m \neq n, T_{m}=\frac{1}{n}$ and $T_{n}=\frac{1}{m}$, then $a-d$,
equals
(A) 0
(B)
(C) $\frac{1}{m n}$
(D) $\frac{1}{m}+\frac{1}{n}$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:02

Problem 145

The sum of the first $\mathrm{n}$ terms of the series $\mathrm{I}^{2}+2 \cdot 2^{2}$ $+3^{2}+2 \cdot 4^{2}+5^{2}+2 \cdot 6^{2}+\ldots$ is $\frac{n(n+1)^{2}}{2}$ when $n$ is
even. When $n$ is odd the sum is $\quad$ [2004]
(A) $\frac{3 n(n+1)}{2}$
(B) $\frac{n^{2}(n+1)}{2}$
(C) $\frac{n(n+1)^{2}}{4}$
(D) $\left[\frac{n(n+1)}{2}\right]^{2}$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:42

Problem 146

If $x=\sum_{n=0}^{\infty} a^{\prime \prime}, y=\sum_{n=0}^{\infty} b^{n}, z=\sum_{n=0}^{\infty} c^{n}$ where $a, b, c$ are
in A.P. and $|a|<1,|b|<1,|c|<1$, then $x, y, z$ are in $|2005|$
(A) G.P.
(B) A.P.
(C) Arithmetic - Geometric Progression
(D) H.P.

Wendi Zhao
Wendi Zhao
Numerade Educator
02:01

Problem 147

Let $a_{1}, a_{2}, a_{3}, \ldots$ be terms of an A.P. If $\frac{a_{1}+a_{2}+\ldots a_{p}}{a_{1}+a_{2} \ldots+a_{q}}$
$=\frac{p^{2}}{q^{2}}, p \neq q$, then $\frac{a_{6}}{a_{21}}$ equals $\quad$ [2006]
(A) $\frac{41}{11}$
(B) $\frac{7}{2}$
(C) $\frac{2}{7}$
(D) $\frac{11}{41}$

Wendi Zhao
Wendi Zhao
Numerade Educator
02:21

Problem 148

If $a_{1}, a_{2}, \ldots, a_{n}$ are in H.P., then the expression $a_{1} a_{2}+$ $a_{2} a_{3}+\ldots+a_{n-1} a_{n}$ is equal to
[2006]
(A) $n\left(a_{1}-a_{n}\right)$
(B) $(n-1)\left(a_{1}-a_{n}\right)$
(C) $n a_{1} a_{n}$
(D) $(n-1) a_{1} a_{n}$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:34

Problem 149

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals
(A) $\frac{1}{2}(1-\sqrt{5})$
(B) $\frac{1}{2} \sqrt{5}$
(C) $\sqrt{5}$
(D) $\frac{1}{2}(\sqrt{5}-1)$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:21

Problem 150

If $p$ and $q$ are positive real numbers such that $p^{2}+q^{2}$ $=1$, then the maximum value of $(p+q)$ is $\underline{\text { [2007 }}$
(A) 2
(B) $1 / 2$
(C) $\frac{1}{\sqrt{2}}$
(D) $\sqrt{2}$

Wendi Zhao
Wendi Zhao
Numerade Educator
01:42

Problem 151

The first two terms of a geometric progression add up to 12 . The sum of the third and the fourth terms is 48 . If the terms of the geometric progression are alternately positive and negative, then the first term is $|2008|$
(A) $-4$
(B) $-12$
(C) 12
(D) 4

Wendi Zhao
Wendi Zhao
Numerade Educator
01:52

Problem 152

The sum to the infinity of the series $1+\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}+\frac{14}{3^{4}}+\ldots \ldots .$ is
(A) 2
(B) $\underline{3}$
(C) $\overline{4}$
(D) 6

Wendi Zhao
Wendi Zhao
Numerade Educator
06:56

Problem 153

A person is to count 4500 currency notes. Let $a_{n}$ denote the number of notes he counts in the $n^{\text {th }}$ minute. If $a_{1}=a_{2}=\ldots \ldots=a_{10}=150$ and $a_{10}, a_{11} \ldots$ are in
A.P. with common difference $-2$, then the time taken by him to count all notes is
(A) 34 minutes
(B) 125 minutes
(C) 135 minutes
(D) 24 minutes

Mohammed Nadhir
Mohammed Nadhir
Numerade Educator
07:03

Problem 154

A man saves Rs. 200 in each of the first three months of his service. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediate preceding month. His total saving from the start of service will be Rs, 11040 after $|\mathbf{2 0 1 1}|$
(A) 19 months
(B) 20 months
(C) 21 months
(D) 18 months

Aflah M
Aflah M
Numerade Educator
03:21

Problem 155

Statement 1: The sum of the series $1+(1+2+4)+$ $(4+6+9)+(9+12+16)+\ldots . .+(361+380+400)$
is 8000 . Statement 2: $\sum_{k=1}^{n}\left(k^{3}=(k-1)^{3}\right)=n^{3}$ for any natural number $n .$
(A) Statement 1 is false, statement 2 is true
(B) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1
(C) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1
(D) Statement 1 is true, statement 2 is false

Wendi Zhao
Wendi Zhao
Numerade Educator
03:59

Problem 156

If 100 times the $100^{\text {th }}$ term of an Arithmetic Progression with non zero common difference equals the 50 times its $50^{\text {th }}$ term, then the $150^{\text {th }}$ term of this
A.P. is |2012|
(A) $-150$
(B) 150 times its $50^{\text {th }}$ term
(C) 150
(D) zero

Aflah M
Aflah M
Numerade Educator
08:28

Problem 157

The sum of first 20 terms of the sequence $0.7,0.77$, $0.777, \ldots$, is
(A) $\frac{7}{9}\left(99-10^{-20}\right)$
(B) $\frac{7}{81}\left(179+10^{-20}\right)$
(C) $\frac{7}{9}\left(99+10^{-20}\right)$
(D) $\frac{7}{81}\left(179-10^{-20}\right)$

Aflah M
Aflah M
Numerade Educator
02:02

Problem 158

Let $\alpha$ and $\beta$ be the roots of equation $p x^{2}+q x+r-0, p \neq 0 .$ If $p, q, r$ are in A.P. and $\frac{1}{\alpha}+\frac{1}{\beta}=4$, then the value of $|\alpha-\beta|$ is
$|2014|$
(A) $\frac{\sqrt{61}}{9}$
(B) $\frac{2 \sqrt{17}}{9}$
(C) $\frac{\sqrt{34}}{9}$
(D) $\frac{2 \sqrt{13}}{9}$

Aman Gupta
Aman Gupta
Numerade Educator
02:14

Problem 159

Three positive numbers form an increasing G.P. If the middle term in this $G . P$ is doubled, the new numbers are in $A . P$. Then the common ratio of the $G . P$ is
(A) $\sqrt{2}+\sqrt{3}$
$\mid 2014]$
(B) $3+\sqrt{2}$
(C) $2-\sqrt{3}$
(D) $2+\sqrt{3}$

Aayush Gupta
Aayush Gupta
Numerade Educator
01:51

Problem 160

If $(10)^{9}+2(11)^{1}(10)^{8}+3(11)^{2}(10)^{7}+\ldots .+10(11)^{9}$
$=k(10)^{9}$ then $k$ is equal to $\quad$ [2014]
(A) $\frac{121}{10}$
(B) $\frac{441}{100}$
(C) 100
(D) 110

Ankur S
Ankur S
Numerade Educator
01:01

Problem 161

The sum of first 9 terms of the series $\frac{1^{3}}{1}+\frac{1^{3}+2^{3}}{1+3}+\frac{1^{3}+2^{3}+3^{3}}{1+3+5}+\ldots . .$ is:
(A) 96
(B) 142
(C) 192
(D) 71

Narayan Hari
Narayan Hari
Numerade Educator
05:56

Problem 162

If the $2^{\text {nd }}, 5^{\text {th }}$ and $9^{\text {th }}$ terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is $[2016]$
(A) $\frac{7}{4}$
(B) $\frac{8}{5}$
(C) $\frac{4}{3}$
(D) 1

Aflah M
Aflah M
Numerade Educator
07:39

Problem 163

If the sum of the first terms of the series $\left(1 \frac{3}{5}\right)^{2}+\left(2 \frac{2}{5}\right)^{2}+\left(3 \frac{1}{5}\right)^{2}+4^{2}+\left(4 \frac{4}{5}\right)^{2}+\ldots$, is $\frac{16}{5} \mathrm{~m}$
then $m$ is equal to [2016]
(A) 99
(B) 102
(C) 101
(D) 100

Aflah M
Aflah M
Numerade Educator