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If $\mathrm{S}(x)=(1+x)+2(1+x)^2+3(1+x)^3+\cdots+60(1+x)^{60}, x \neq 0$, and $(60)^2 \mathrm{~S}(60)=\mathrm{a}(\mathrm{b})^{\mathrm{b}}+\mathrm{b}$, where $a, b \in N$, then $(a+b)$ equal to ______
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1189 Upvotes
Verified Answer
The correct answer is:
3660
$\begin{aligned}
S(x)= & (1+x)+2(1+x)^2+3(1+x)^3+. .+60(1+x)^{60} \\
(1+x) S= & (1+x)^2+\ldots \ldots . \quad 59(1+x)^{60}+60(1+x)^{61} \\
& -x S=\frac{(1+x)(1+x)^{60}-1}{x}-60(1+x)^{61}
\end{aligned}$
Put $\mathrm{x}=60$
$-60 \mathrm{~S}=\frac{61\left((61)^{60}-1\right)}{60}-60(61)^{61}$
on solving 3660
S(x)= & (1+x)+2(1+x)^2+3(1+x)^3+. .+60(1+x)^{60} \\
(1+x) S= & (1+x)^2+\ldots \ldots . \quad 59(1+x)^{60}+60(1+x)^{61} \\
& -x S=\frac{(1+x)(1+x)^{60}-1}{x}-60(1+x)^{61}
\end{aligned}$
Put $\mathrm{x}=60$
$-60 \mathrm{~S}=\frac{61\left((61)^{60}-1\right)}{60}-60(61)^{61}$
on solving 3660
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