Question

If the coefficients of 2^nd, 3^rd and the 4^th terms in the expansion of (1 + x)ⁿ are in A.P., then the value of n is
(a) 2
(b) 7
(c) 11
(d) 14

Answer 1

Given coefficient of 2^nd, 3^rd and 4^th terms in the expansion (1 + x)ⁿ are A.P
Since coefficient of (r + 1)^th terms is ⁿC_r
∴ 2^nd, 3^rd and 4^th coefficient are such that 2ⁿC_r = ⁿC₁ + ⁿC₃
$$\begin{gathered} \mathrm{i}.\mathrm{e}.2\times \frac{n(n-1)}{2}=n+\frac{n(n-1)(n-2)}{6} \\ \mathrm{i}.\mathrm{e}.n(n-1)=n\left[1+\frac{\left(n-1\right)(n-2)}{6}\right] \\ \mathrm{i}.\mathrm{e}.n-1=1+\frac{\left(n-1\right)(n-2)}{6} \\ \mathrm{i}.\mathrm{e}.6(n-1)=6+n^2-3n+2 \\ \mathrm{i}.\mathrm{e}.n^2-9n+14=0 \\ \mathrm{i}.\mathrm{e}.n^2-7n-2n+14=0 \\ \mathrm{i}.\mathrm{e}.n(n-7)-2(n-7)=0 \\ \mathrm{i}.\mathrm{e}.n=7\left(\because \mathit{}n=2\mathrm{is}\mathrm{not}\mathrm{possible}\mathrm{since}\mathrm{for}\mathit{}n=2,3\mathrm{rd}\mathrm{and}4\mathrm{th}\mathrm{terms}\mathrm{will}\mathrm{be}0\right) \end{gathered}$$

Hence, the correct answer is option B.