If the sum of the coefficients in the expansion of (1 + 2x)^n is 6561, the greatest term in the expansion for x = 1/2 is

Sol. sum of the coefficient in the expansion of (1 + 2x)ⁿ = 6561

⇒ (1 + 2x)ⁿ = 6561, when x = 1

⇒ 3ⁿ = 6561

⇒ 3ⁿ = 3⁸

⇒ n = 8

$\large \frac{T_{r+1}}{T_r} = \frac{8_{C_r} (2x)^r }{8_{C_{r-1}} (2x)^{r-1}}  = \frac{9-r}{r} \times 2x $

Since x = 1/2 ;

$\large \frac{T_{r+1}}{T_r} = \frac{9-r}{r}  $

$\large \frac{T_{r+1}}{T_r} > 1 \Rightarrow  \frac{9-r}{r} >1  $

9-r > r  ⇒ 2r < 9 ⇒ r < 4.5

Hence, 5^th term is the greatest term.

Hence (B) is the correct answer.