Question

Consider the following two statements :

Statement-I : The sum of series $1+(1+2+4)+(4+6+9)+(9+12+16)+\dots +(361+380+400)$ is $8000$.

Statement-II : $\sum _{k=1}^n\left[k^3-{\left(k-1\right)}^3\right]=n^3$.

Then which of one of the following choices is correct?

• A. Statement-I is incorrect and Statement-II is correct.
• B. Both Statement-I and Statement-II are correct and Statement-II is a correct explanation for Statement-I
• C. Both Statement-I and Statement-II are correct and Statement-II is not a correct explanation for Statement-I.
• D. Statement-I is correct but Statement-II is incorrect.

Answer 1

The correct option is B

Both Statement-I and Statement-II are correct and Statement-II is a correct explanation for Statement-I

Explanation for the correct option :

Step-1 : Evaluation of $\sum _{k=1}^n\left[k^3-{\left(k-1\right)}^3\right]$

Consider the Statement-II

$$\begin{gathered} \sum _{k=1}^n\left[k^3-{\left(k-1\right)}^3\right]=\left(1^3-0^3\right)+\left(2^3-1^3\right)+\left(3^3-2^3\right)+\dots +\left[n^3-{\left(n-1\right)}^3\right] \\ =n^3 \end{gathered}$$

Then for $n=20$, we get

$$\begin{gathered} \left(1^3-0^3\right)+\left(2^3-1^3\right)+\left(3^3-2^3\right)+\dots +\left({20}^3-{19}^3\right)=\sum _{k=1}^{20}\left[k^3-{\left(k-1\right)}^3\right] \\ ={20}^3 \\ =8000 \end{gathered}$$

Step-2 : Evaluation of the series in Statement-I i.e. $1+\left(1+2+4\right)+\left(4+6+9\right)+\dots +\left(361+380+400\right)$.

Here, we shall use the fact that $a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)$

Then

$$\begin{gathered} \left(1^3-0^3\right)+\left(2^3-1^3\right)+\left(3^3-2^3\right)+\dots +\left({20}^3-{19}^3\right)=1+\left(2-1\right)\left(2^2+2.1+1\right)+\left(3-2\right)\left(3^2+3.2+2^2\right)+\dots +\left(20-19\right)\left({20}^2+20.19+{19}^2\right) \\ =1+\left(1+2+4\right)+\left(4+6+9\right)+\dots +\left(361+380+400\right) \end{gathered}$$

Step-3 : Evaluation of the series in Statement-I i.e. $1+\left(1+2+4\right)+\left(4+6+9\right)+\dots +\left(361+380+400\right)$

From Step-1 and Step-2, we can find that

$1+\left(1+2+4\right)+\left(4+6+9\right)+\dots +\left(361+380+400\right)=8000$.

Therefore, both Statement-I and Statement-II are correct and Statement-II is the correct explanation for Statement-I

Hence, option (B) is the correct answer.