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

Answer 1

Answer: (B)

Let $T_{2-1},T_r,T_{r+1}$ be the three successive terms of $(1+x{)}^n$
$\Rightarrow {}^n{C}_r:{}^n{C}_{r-1}:{}^n{C}_r=1:7:42$
$\Rightarrow \frac{{}^n{C}_{r-1}}{{}^n{C}_{r-2}}=7;\frac{{}^n{C}_r}{{}^n{C}_{r-1}}=\frac{42}{7}=6$ $\Rightarrow \frac{n-r+2}{r-1}=7;\frac{n-r+1}{r}=6$
$\Rightarrow r=8,n=55$
$\Rightarrow T_{r-1}=T_7$
$\Rightarrow$ First of the three given terms will be $7^{th}$