First slide

Binomial theorem for positive integral Index

Question

$The sum of coefficients of the expansion {(\frac{1}{x} + 2 x)}^{n} is 6561 . The coefficient of term independent of x is$

Moderate

A

$16 \cdot {}^{8}C_{4}$
B

$8 \cdot {}^{8}C_{4}$
C

${}^{8}C_{5}$
D

$n o t e r m i n d e p e n d e n t o f x$

Solution

$Tthe sum of the oefficients of the expansion {(\frac{1}{x} + 2 x)}^{n} = 6561 Therefore (1 + 2)^{n} = 3^{8} \Rightarrow 3^{n} = 3^{8} \Rightarrow n = 8 Let (r + 1) th term is independent of x . Therefore, T_{r + 1} = {}^{8}C_{r} (\frac{1}{x}) (2 x)^{8 - r} = {}^{8}C {{}_{r} 2 {}^{8 - r} x}^{8 - 2 r} Since this term is independent of x, then 8 - 2 r = 0 \Rightarrow r = 4 Therefore, coefficient of independent term is T_{5} = {}^{8}C_{4} \cdot 2^{4} = 16 \cdot {}^{8}C_{4}$

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