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

The correct option is B $7$th

Let the three consucutive terms will be $r^{\text{th}},(r+1{)}^{\text{th}},(r+2{)}^{\text{th}}$.

${}^n{C}_{r-1}:{}^n{C}_r:{}^n{C}_{r+1}=1:7:42\frac{r}{n-r+1}=\frac{1}{7}\Rightarrow n=8r-1...\left(1\right)\frac{r+1}{n-r}=\frac{7}{42}=\frac{1}{6}\Rightarrow n=7r+6...\left(2\right)\text{Solving eqn(1) and (2), we get,}\Rightarrow r=7$