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Section 3

Series

Find the $20^{\text {th }}$ and $n^{\text {th }}$ terms of the G.P. $\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \ldots$

Find the $12^{\text {th }}$ term of a G.P. whose $8^{\text {th }}$ term is 192 and the common ratio is 2 .

The $5^{\text {th }}, 8^{\text {th }}$ and $11^{\text {th }}$ terms of a G.P. are $p, q$ and $s$, respectively. Show that $q^{2}=p s$.

The $4^{\text {th }}$ term of a G.P. is square of its second term, and the first term is $-3$. Determine its $7^{\text {th }}$ term.

Which term of the following sequences:(a) $\quad 2,2 \sqrt{2}, 4, \ldots$ is $128 ?$(b) $\sqrt{3}, 3,3 \sqrt{3}, \ldots$ is $729 ?$(c) $\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots$ is $\frac{1}{19683}$ ?

For what values of $x$, the numbers $-\frac{2}{7}, x,-\frac{7}{2}$ are in G.P.?

Find the sum to indicated number of terms in each of the geometric progressions in Exercises 7 to 10 :$0.15,0.015,0.0015, \ldots 20$ terms.

Find the sum to indicated number of terms in each of the geometric progressions in Exercises 7 to 10 :$\sqrt{7}, \sqrt{21}, 3 \sqrt{7}, \ldots n$ terms.

Find the sum to indicated number of terms in each of the geometric progressions in Exercises 7 to 10 :$1,-a, a^{2},-a^{3}, \ldots n$ terms (if $\left.a \neq-1\right)$

Find the sum to indicated number of terms in each of the geometric progressions in Exercises 7 to 10 :$x^{3}, x^{5}, x^{7}, \ldots n$ terms (if $\left.x \neq \pm 1\right)$.

Evaluate $\sum_{k=1}^{11}\left(2+3^{k}\right)$.

The sum of first three terms of a G.P. is $\frac{39}{10}$ and their product is $1 .$ Find the common ratio and the terms.

How many terms of G.P. $3,3^{2}, 3^{3}, \ldots$ are needed to give the sum $120 ?$

The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to $n$ terms of the G.P.

Given a G.P. with $a=729$ and $7^{\text {th }}$ term 64 , determine $\mathrm{S}_{7}$.

Find a G.P. for which sum of the first two terms is $-4$ and the fifth term is 4 times the third term.

If the $4^{\text {th }}, 10^{\text {th }}$ and $16^{\text {th }}$ terms of a G.P. are $x, y$ and $z$, respectively. Prove that $x$, $y, z$ are in G.P.

Find the sum to $n$ terms of the sequence, $8,88,888,8888 \ldots$.

Find the sum of the products of the corresponding terms of the sequences $2,4,8$,$16,32$ and $128,32,8,2, \frac{1}{2}$.

Show that the products of the corresponding terms of the sequences $a, a r, a r^{2}$, $\ldots a r^{n-1}$ and $\mathrm{A}, \mathrm{AR}, \mathrm{AR}^{2}, \ldots \mathrm{AR}^{n-1}$ form a G.P, and find the common ratio.

Find four numbers forming a geometric progression in which the third term is greater than the first term by 9 , and the second term is greater than the $4^{\text {th }}$ by 18 .

If the $p^{\text {th }}, q^{\text {th }}$ and $r^{\text {? }}$ terms of a G.P. are $a, b$ and $c$, respectively. Prove that$$a^{q-r} b^{r-p} c^{P-q}=1.$$

If the first and the $n^{\text {th }}$ term of a G.P. are $a$ and $b$, respectively, and if $\mathrm{P}$ is the product of $n$ terms, prove that $\mathrm{P}^{2}=(a b)^{n}$.

Show that the ratio of the sum of first $n$ terms of a G.P. to the sum of terms from $(n+1)^{\mathrm{th}}$ to $(2 n)^{\mathrm{th}}$ term is $\frac{1}{r^{n}}$.

If $a, b, c$ and $d$ are in G.P. show that $\left(a^{2}+b^{2}+c^{2}\right)\left(b^{2}+c^{2}+d^{2}\right)=(a b+b c+c d)^{2}$.

Insert two numbers between 3 and 81 so that the resulting sequence is G.P.

Find the value of $n$ so that $\frac{a+b}{a^{n}+b^{n}}$ may be the geometric mean between $a$ and $b$.

The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio $(3+2 \sqrt{2}):(3-2 \sqrt{2})$.

If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are $\mathrm{A} \pm \sqrt{(\mathrm{A}+\mathrm{G})(\mathrm{A}-\mathrm{G})}$.

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of $2^{\text {nd }}$ hour, $4^{\text {th }}$ hour and $n^{\text {th }}$ hour ?

What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of $10 \%$ compounded annually?

If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the quadratic equation.