QJEEM16Mac9f9: JEE Main 2016 Mathematics - Class 11 JEEM16Mac9f9 Solution: For x R , x -1, if (1 + x) 2016 + x(1 + x) 2015 + x 2 (1 + x)…

Question

Verified Solution • 2025–26 Session moderate

For x

\in

R, x

\ne

-1,

if (1 + x)²⁰¹⁶ + x(1 + x)²⁰¹⁵ + x²(1 + x)²⁰¹⁴ + . . . . + x²⁰¹⁶ =

\sum\limits_{i = 0}^{2016} {{a_i}} \,{x^i},\,\,

then a₁₇ is equal to :

Accepted Answer

Assume,

P = (1 + x)²⁰¹⁶ + x(1 + x)²⁰¹⁵ + . . . . .+ x²⁰¹⁵ . (1 + x) + x²⁰¹⁶ . . . . .(1)

Multiply this with

\left( {{x \over {1 + x}}} \right),

\left( {{x \over {1 + x}}} \right)P =

x(1 + x)²⁰¹⁵ + x²(1 + x)²⁰¹⁴ +

. . . . . . + x²⁰¹⁶ +

{{{x^{2017}}} \over {1 + x}}

. . . . . (2)

Performing (1)

-

(2), we get

{P \over {1 + x}} =

(1 + x)²⁰¹⁶

-

{{{x^{2017}}} \over {1 + x}}

\Rightarrow

P = (1 + x)²⁰¹⁷

-

x²⁰¹⁷

\therefore

a₁₇ = coefficient of x¹⁷

=

²⁰¹⁷C₁₇

=

{{2017!} \over {17!\,\,2000!}}

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