JEE Main 2024 6 April Evening Shift
Exam JEE Main
Year 2024
Date 6 April
Shift Evening Shift
Subject Mathematics
Chapter Sequences and Series JEE Main 2024 · 6 April · Evening Shift · Mathematics · Sequences and Series
If S(x)=(1+x)+2(1+x)^2+3(1+x)^3+ +60(1+x)^ 60 , x ≠ 0 , and (60)^2...
If
S ( x ) = ( 1 + x ) + 2 ( 1 + x ) 2 + 3 ( 1 + x ) 3 + ⋯ + 60 ( 1 + x ) 60 , x ≠ 0 \mathrm{S}(x)=(1+x)+2(1+x)^2+3(1+x)^3+\cdots+60(1+x)^{60}, x \neq 0 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 (60)^2 \mathrm{~S}(60)=\mathrm{a}(\mathrm{b})^{\mathrm{b}}+\mathrm{b} ( 60 ) 2 S ( 60 ) = a ( b ) b + b , where
a , b ∈ N a, b \in N a , b ∈ N , then
( a + b ) (a+b) ( a + b ) equal to ______
Step-by-step solution
S ( x ) = ( 1 + x ) + 2 ( 1 + x ) 2 + 3 ( 1 + x ) 3 + . . + 60 ( 1 + x ) 60 ( 1 + x ) S = ( 1 + x ) 2 + … … . 59 ( 1 + x ) 60 + 60 ( 1 + x ) 61 − x S = ( 1 + x ) ( 1 + x ) 60 − 1 x − 60 ( 1 + x ) 61 \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} S ( x ) = ( 1 + x ) S = ( 1 + x ) + 2 ( 1 + x ) 2 + 3 ( 1 + x ) 3 + .. + 60 ( 1 + x ) 60 ( 1 + x ) 2 + …… . 59 ( 1 + x ) 60 + 60 ( 1 + x ) 61 − x S = x ( 1 + x ) ( 1 + x ) 60 − 1 − 60 ( 1 + x ) 61 Put
x = 60 − 60 S = 61 ( ( 61 ) 60 − 1 ) 60 − 60 ( 61 ) 61 \mathrm{x}=60 -60 \mathrm{~S}=\frac{61\left((61)^{60}-1\right)}{60}-60(61)^{61} x = 60 − 60 S = 60 61 ( ( 61 ) 60 − 1 ) − 60 ( 61 ) 61
on solving 3660
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