Question

If the coefficients of the three successive terms in the binomial expansion of $(1+x{)}^n$ are in the ratio $1:7:42$, then the first of these terms in the expansion is :

• A. $7^{th}$
• B. $6^{th}$
• C. $8^{th}$
• D. $9^{th}$

Answer 1

The correct option is A $7^{th}$

Let the three consecutive terms in the expansion of $(1+x{)}^n$ are $(r1{)}^{th},r^{th}$ and $(r+1{)}^{th}$ terms.
$T_{r-1}=T_{r-2+1}{=}^n{C}_{r-2}{x}^{r-1}$
$T_r=T_{r-1+1}{=}^n{C}_{r-1}{x}^r$
$T_{r+1}{=}^n{C}_r{x}^r$
Given, $\frac{{}^n{C}_{r-2}}{{}^n{C}_{r-1}}=\frac{1}{7}$
$\Rightarrow n-8r+9=0$ .....(1)
$\frac{{}^n{C}_{r-1}}{{}^n{C}_r}=\frac{1}{6}$
$\Rightarrow n-7r+1=0$ .....(2)
Solving (1) and (2), we get
$n=55,r=8$
So, first term will be $T_7$.