Question

Assertion :1 The sum of the series$1+(1+2+4)+(4+6+9)+(9+12+16)+.+(361+380+400)$ is $8000$ Reason: ${\sum }_{t=1}^n(k^3-{(k-1)}^3)=n^3$ for any natural number $n$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Since,$\sum _{k=1}^n{k}^2=\frac{1}{6}n(n+1)(2n+1)$

$\sum _{k=1}^{n}k=\frac{1}{2}n(n+1)$ & $\sum _{k=1}^{n}1=n$

general term of the series is,

$T_n=(n-1{)}^2+n(n-1)+n^2=3n^2-3n+1$

$=[n^3-(n-1{)}^3]=\frac{n}{2}(n+1\left)\right(2n+1)-\frac{3}{2}n(n+1)+n$

when $n=20$, then,

$\sum _{n=1}^{20}{T}_n=\frac{20}{2}(20+1)(40+1)-\frac{3}{2}.20.21+20$

$=8610-630+20=8000$

So option (A) is correct.