Question

The coefficients of three consecutive terms in the expansion of $(1 + a)^n$ are in the ratio $1 : 7 : 42$. Find $n$.

• A. $45$
• B. $55$
• C. $40$
• D. $50$

Answer 1

Answer: (B)

$55$

Answer 2

Suppose the three consecutive terms in the expansion of $(1 + a)^n$ are $(r - 1)^{th}$, $r^{th}$ and $(r + 1)^{th}$ terms. Coefficient of $T_{(r-2)+1} = \,^nC_{r-2}$ Coefficient of $T_{(r- 1)+1} = \,^nC_{r-1}$ Coefficient of $T_r+1 = ^nC_r$ Since the coefficients are in the ratio $1 : 7 : 42$, so we have, $\frac{^{n}C_{r-2}}{^{n}C_{r-1}} = \frac{1}{7}$, i.e., $n -8r+9 = 0\quad\ldots\left(1\right)$ and $\frac{^{n}C_{r-1}}{^{n}C_{r}} = \frac{7}{42}$, i.e., $n -7r+1 = 0\quad \ldots \left(2\right)$ Solving $\left(1\right)$ and $\left(2\right)$ we get, $n = 55$.