In the binomial expansion of (√ x +(1/2 ⋅ √[4] x )) n , the first three coefficients form an arithmetic progression, then sum of coefficients of all the terms is

Question

In the binomial expansion of (√ x +(1/2 ⋅ √[4] x )) n , the first three coefficients form an arithmetic progression, then sum of coefficients of all the terms is (A) ((3/2))5 (B) ((3/2))6 (C) ((3/2))7 (D) ((3/2))8

Tardigrade Admin · Accepted Answer

n C 0+ n C 2 (1/22)=2 n C 1 (1/2) ⇒ 1+( n ( n -1)/8)= n =( n 2- n +8/8) ⇒ n 2-9 n +8=0 ⇒ n =1,8 n =1 text (rejected) ∴ text sum of coefficient =(1+(1/2))8=((3/2))8

Q. In the binomial expansion of $(x + \frac{1}{2 \cdot 4 x})^{n}$ , the first three coefficients form an arithmetic progression, then sum of coefficients of all the terms is

$(\frac{3}{2})^{5}$

$(\frac{3}{2})^{6}$

$(\frac{3}{2})^{7}$

$(\frac{3}{2})^{8}$

$^{n} C_{0} +^{n} C_{2} \frac{1}{2 ^{2}} = 2^{n} C_{1} \frac{1}{2}$
$\Rightarrow 1 + \frac{n ( n - 1 )}{8} = n = \frac{n ^{2} - n + 8}{8}$
$\Rightarrow n^{2} - 9 n + 8 = 0 \Rightarrow n = 1, 8 n = 1 (rejected)$
$∴ sum of coefficient = (1 + \frac{1}{2})^{8} = (\frac{3}{2})^{8}$

Q. In the binomial expansion of (x​+2⋅4x​1​)n, the first three coefficients form an arithmetic progression, then sum of coefficients of all the terms is

(23​)5

(23​)6

(23​)7

(23​)8