Question

Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R
Assertion (A) : (1) \(+(1+2+4)+(4+6+9) +(9+12+16)+\ldots+(81+90+100)=1000\) 

Reason (R) : \(\displaystyle \sum_{r=1}^{n}\left(r^{3}-[r-1]^{3}\right)=n^{3}\) for any natural number \(n\)

In the light of the above statements, choose the correct answer from the options given below

• A. Both (A) and (R) are true and (R) is the coll ect explanation of (A)
• B. Both (A) and (R) are tiue but (R) is not the collect explanation of (A)
• C. (A) is tme but (R) is false
• D. (A) is false but (R) is true

Answer 1

The Correct Option is A

Since,
$1+(1+2+4)+(4+6+9)+(9+12+16) +...+(81+90+100)$
$= 1+\left(1^{2}+(2 \times 1)+2^{2}\right)+\left(2^{2}+(2 \times 3)+3^{2}\right)+\left(3^{2}\right.$
$\left.+(3 \times 4)+4^{2}\right)+\ldots+\left(9^{2}+(9 \times 10)+10^{2}\right)$
$=\displaystyle \sum_{r=1}^{10}\left[(r-1)^{2}+r(r-1)+r^{2}\right]$
$=\displaystyle \sum_{r=1}^{10}[r-(r-1)]\left[(r-1)^{2}+r(r-1)+r^{2}\right]$
$=\displaystyle \sum_{r=1}^{10}\left[r^{3}-(r-1)^{3}\right]$
$=\left(1^{3}-0^{3}\right)+\left(2^{3}-1^{3}\right)+\left(3^{3}-2^{3}\right)+\ldots+\left(10^{3}-9^{3}\right)$
$= 10^{3}-0^{3}=1000$
So, both (A) and $( R )$ are true and $( R )$ is the correct explanation of $(A)$.