Question

If the coefficients of $r-1^{th}$, $r^{th}$ and $r+1^{th}$ terms in $(x+1{)}^n$ are in the ratio $1:3:5$, then $n=$

• A. $5$
• B. $6$
• C. $7$
• D. $8$

Answer 1

The correct option is C $7$

It is given that
$\frac{{}^n{C}_{r-2}}{{}^n{C}_{r-1}}=\frac{1}{3}$

$3{}^n{C}_{r-2}={}^n{C}_{r-1}$

$\frac{3}{n+2-r}=\frac{1}{r-1}$

$3r-3=n+2-r$

$4r=n+5$ ...(i)

And
$\frac{{}^n{C}_{r-1}}{{}^n{C}_r}=\frac{3}{5}$

$5{}^n{C}_{r-1}=3{}^n{C}_r$

$\frac{5}{n+1-r}=\frac{3}{r}$

$5r=3n+3-3r$

$8r=3n+3$ ...(ii)
Subtracting Eq(i) from 2.Eq(ii) we get.
$-n+7=0$

$n=7$