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 ________.
Correct Answer: 3660
Solution and Explanation
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:
\[S = 3660\]
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