The sum of coefficients of the expansion \((\frac{1}{x}+2x)^n\) is 656

1.	The sum of coefficients of the expansion \((\frac{1}{x}+2x)^n\) is 6561. The coefficient of term independent of x is(a) 16. 8C4 (b) 8C4 (c) 8C5 (d) None of these
Answer» Answer : (a) 16. ⁸C₄ Sum of the coefficients of the expansion \((\frac{1}{x} +2x)^n\)_{= 6561} Putting x = 1, (1 + 2)ⁿ = 6561 ⇒ 3ⁿ= 3⁸ ⇒ n = 8 \(\therefore\) T_{r + 1} in the expansion of \((\frac{1}{x} +2x)^8\) = ⁸C_r \((\frac{1}{x})^{8-r}\) (2x)^r = ⁸C_r 2^rx^2r-8 Since this term is independent of x, 2r – 8 = 0 ⇒ r = 4. ∴ Reqd. term = ⁸C₄ . 2⁴ = 16 . ⁸C₄.

The sum of coefficients of the expansion \((\frac{1}{x}+2x)^n\) is 6561. The coefficient of term independent of x is(a) 16. 8C4 (b) 8C4 (c) 8C5 (d) None of these

Answer»

Answer : (a) 16. ⁸C₄

Sum of the coefficients of the expansion \((\frac{1}{x} +2x)^n\)_{= 6561}

Putting x = 1, (1 + 2)ⁿ = 6561 ⇒ 3ⁿ= 3⁸

⇒ n = 8

\(\therefore\) T_{r + 1} in the expansion of \((\frac{1}{x} +2x)^8\)

= ⁸C_r \((\frac{1}{x})^{8-r}\) (2x)^r = ⁸C_r 2^rx^2r-8

Since this term is independent of x, 2r – 8 = 0 ⇒ r = 4.

∴ Reqd. term = ⁸C₄ . 2⁴

= 16 . ⁸C₄.