Question

The coeffecients of three consecutive term in the expansion of ${(1+x)}^n$ are in the ratio $1:7:42$, then find the value of $n$

• A. $n=54$
• B. $n=55$
• C. $n=53$
• D. $n=56$

Answer 1

Answer: (B)

$n=55$

It is given that
$\frac{{}^n{C}_{r-2}}{{}^n{C}_{r-1}}=\frac{1}{7}$
$7{}^n{C}_{r-2}={}^n{C}_{r-1}$
$\frac{7}{n+2-r}=\frac{1}{r-1}$
$7r-7=n+2-r$
$8r=n+9$ ...(i)
And
$\frac{{}^n{C}_{r-1}}{{}^n{C}_r}=\frac{7}{42}$
$6{}^n{C}_{r-1}={}^n{C}_r$
$\frac{6}{n+1-r}=\frac{1}{r}$
$6r=n+1-r$
$7r=n+1$ ...(ii)
Subtractin Eq(i) from 2.Eq(ii) we get.
$r=8$
Hence
$n=55$