Question

If ${\mathrm{S}}_{\mathrm{n}}=1^3+2^3+\dots ..+{\mathrm{n}}^3$ and ${\mathrm{T}}_{\mathrm{n}}=1+2+\dots ..+\mathrm{n}$, then

• A. ${\mathrm{S}}_{\mathrm{n}}={\mathrm{T}}_{\mathrm{n}}$
• B. ${\mathrm{S}}_{\mathrm{n}}={\mathrm{T}}_{\mathrm{n}}^4$
• C. ${\mathrm{S}}_{\mathrm{n}}={\mathrm{T}}_{\mathrm{n}}^2$
• D. $S_n=T_n^3$

Answer 1

The correct option is C

${\mathrm{S}}_{\mathrm{n}}={\mathrm{T}}_{\mathrm{n}}^2$

Explanation for the correct option

Given, ${\mathrm{S}}_{\mathrm{n}}=1^3+2^3+\dots ..+{\mathrm{n}}^3$ and ${\mathrm{T}}_{\mathrm{n}}=1+2+\dots ..+\mathrm{n}$.

$$\begin{gathered} {\mathrm{S}}_{\mathrm{n}}=\sum {\mathrm{n}}^3 \\ {\mathrm{T}}_{\mathrm{n}}=\sum \mathrm{n} \end{gathered}$$

We know that,

$$\begin{gathered} \Rightarrow {\mathrm{S}}_{\mathrm{n}}=\sum {\mathrm{n}}^3 \\ ={\left[\frac{\mathrm{n}(\mathrm{n}+1)}{2}\right]}^2 \\ \Rightarrow {\mathrm{S}}_{\mathrm{n}}={\left\{\sum \mathrm{n}\right\}}^2 \\ ={\mathrm{T}}_{\mathrm{n}}^2 \end{gathered}$$

Hence, option C is correct.