Question

If S_n = 1³ + 2³ + ..... + n³ and T_n = 1 + 2 + 3 + ..... + n, then

(a) S_n = \(T^2_n\)

(b) S_n = \(T^3_n\)

(c) \(S_n^2\) = T_n 

(d) \(S_3^2\) = T_n

Answer 1

(a) S_n = \(T_n^2\)

Sn = ∑n³

= \(\frac{n^2(n+1)^2}{4}\) = \(\big(\frac{n(n+1)}{2}\big)^2\)                ....(i)

T_n = \(\frac{n(n+1)}{2}\)                   ......(ii)

∴ From (i) and (ii) 

S_n = \(T_n^2\)