Question

If the coefficients of three consecutive terms in the expansion of $\left(1 + x\right)^{n}$ are in the ratio $1:7:42,$ then the value of $n$ is equal to

• A. $49$
• B. $50$
• C. $55$
• D. $56$

Answer 1

Answer: (C)

For $T_{r + 1}= \,{}^{n}C_{r}x^{r},$ coefficient $ \rightarrow \,{}^{n}C_{r}$
$ \,{}^{n}C_{r - 1}: \,{}^{n}C_{r}: \,{}^{n}C_{r + 1}=1:7:42$
$\Rightarrow \frac{ \,{}^{n} C_{r}}{ \,{}^{n} C_{r - 1}}=\frac{7}{1}$ and $\frac{ \,{}^{n} C_{r + 1}}{ \,{}^{n} C_{r}}=\frac{42}{7}$
$\Rightarrow \frac{n - r + 1}{r}=7$ and $\frac{n - r}{r + 1}=6$
$\Rightarrow n-r+1=7r$ and $n-r=6r+6$
$\Rightarrow n=8r-1$ and $ \, n=7r+6$
$\Rightarrow 8r-1=7r+6\Rightarrow r=7$
$\Rightarrow n=55$