If S(x) = ( 1 + x) + 2(1 + x)^2 + 3(1 + x)^3 + .....+ 60(1 + x)^60, x ≠ 0, and

Question

If  \(S(x)=(1+x)+2(1+x)^{2}+3(1+x)^{3}+\ldots\)\(+60(1+x)^{60}, x \neq 0\) and \((60)^{2}\, S(60)=a(b)^{b}+b\), where a, b ∈ N, then (a + b) equal to ______

Answer 1

Correct answer is: 3660

\(S(x)=(1+x)+2(1+x)^{2}+3(1+x)^{3}+\ldots+60(1+x)^{60}\)

\((1+x) S=(1+x)^{2}+\ldots + 59(1+x)^{60}+60(1+x)^{61}\)

\(-x S=\frac{(1+x)(1+x)^{60}-1}{x}-60(1+x)^{61}\)

Put x = 60

\( -60S = \frac{61((61)^{60}-1)}{60}-60(61)^{61}\)

On solving,

a + b = 3660