Question

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

• 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.

Answer 1

Answer: (B)

$T_{n} = \left(n -1\right)^{2} + \left(n - 1\right)n + n^{2} = \frac{\left(\left(n-1\right)^{3}-n^{3}\right)}{\left(n-1\right)-n} = n^{3} - \left(n - 1\right)^{3}$
$T_{1} = 1^{3} - 0^{3}$
$T_{2} = 2^{3} - 1^{3}$
$⋮$
$T_{20} = 20^{3} - 19^{3}$
$S_{20} = 20^{3} - 0^{3} = 8000$