If S ( x ) = ( 1 + x ) + 2 ( 1 + x ) 2 + 3 ( 1 + x ) 3 + … + 60 ( 1 + x ) 60 , x ? 0 and ( 60 ) 2 S ( 60 ) = a ( b ) b + b , where a , b ? N , then ( a + b ) is equal to _______. [2024]

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Solution

Q.
If $S (x) = (1 + x) + 2 {(1 + x)}^{2} + 3 {(1 + x)}^{3} + \dots + 60 {(1 + x)}^{60}$ , $x \neq 0$ and ${(60)}^{2} S (60) = a {(b)}^{b} + b,$ where $a, b \in N,$ then $(a + b)$ is equal to _______. [2024]

Ans.
(3660)

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

$\Rightarrow x^{2} S (x) = 60 x \cdot {(1 + x)}^{61} - (1 + x) ({(1 + x)}^{60} - 1)$

For $x = 60,$ we have

${(60)}^{2} S (60) = 3600 {(61)}^{61} - 61 ({(61)}^{60} - 1)$

$= {(61)}^{61} [3600 - 1] + 61 = 3599 {(61)}^{61} + 61$

$\Rightarrow a = 3599, b = 61$

$∴ a + b = 3599 + 61 = 3660$

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