Question

If $ S_n = 1^3 + 2^3 + \ldots + n^3 $ and $ T_n = 1 + 2 + \ldots + n $, then

• A. \( S_n = T_n^3 \)
• B. \( S_n = T_n^5 \)
• C. \( S_n = T_n^4 \)
• D. \( S_n = T_n^2 \)

Hint:

Memorize the identity \( \sum_{k=1}^n k^3 = \left( \sum_{k=1}^n k \right)^2 \). It’s useful in series and binomial problems.

Answer 1

The Correct Option is D

We know: \[ T_n = \sum_{k=1}^n k = \frac{n(n+1)}{2} \] \[ S_n = \sum_{k=1}^n k^3 = \left( \frac{n(n+1)}{2} \right)^2 \] Thus, \[ S_n = T_n^2 \]