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 $a, b \in \mathbb{N}$, then $(a + b)$ is equal to ________.

Answer 1

Starting with the series:
$S(x) = (1 + x) + 2(1 + x)^2 + 3(1 + x)^3 + \dots + 60(1 + x)^{60}$
Multiplying both sides by $(1 + x)$, we get:
$(1 + x)S = (1 + x) + 2(1 + x)^2 + 3(1 + x)^3 + \dots + 60(1 + x)^{61}$
Now, subtracting $S$ from $(1 + x)S$, we obtain:
$-xS = \frac{(1 + x)(1 + x)^{60} - 1}{x} - 60(1 + x)^{61}$
Now, put $x = 60$:
$-60S = \frac{61((61)^{60} - 1)}{60} - 60 \cdot (61)^{61}$
Solving this equation gives:
$S = 3660$