Statement-1: The sum of the series 1+(1+2+4)+ (4+6+9)+(9+12+16)+…+(361+380+400) is 8000 . S

Name: Problem 42, Chapter 10: A Complete Resource Book in Mathematics for JEE Main
Uploaded: 2021-09-18T11:11:55-08:00
Duration: 3 min 21 s
Channel: Wendi Zhao

step 1 For this one in the beginning, we can write it as 0 plus 0 plus 1. And now let's look at each pe

Question

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.

   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.

A Complete Resource Book in Mathematics for JEE Main

Dinesh Khattar 2019 Edition

Chapter 10, Problem 42

Instant Answer

Solved by Expert Wendi Zhao

09/18/2021

Step 1

Step 1: We can rewrite the series as follows: $0+0+1+(1^2+1*0)+(2^2+2*1)+(3^2+3*2)+\ldots+(20^2+20*19)$ Show more…

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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.

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Key Concepts

Telescoping Series

A telescoping series is one in which many terms cancel out successively, leaving only a relatively small number of terms from the beginning and end of the sum. This cancellation greatly simplifies the evaluation of the sum and is often used in proofs and derivations of closed?form summation formulas.

Difference of Consecutive Cubes

The difference of consecutive cubes, such as k³ - (k-1)³, is a special algebraic form whose expansion leads to a telescoping behavior when summed over a sequence of natural numbers. This property is utilized to derive neat formulas for sums and to simplify expressions that might otherwise appear complicated.

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