Question

The coefficients of the (r-1), r th and (r+1)th terms in the expansion of $(x+n{)}^n$ are in the ratio 1:3:5. Find n and r.

Answer 1

We know that coefficients of (r-1)th, r th and (r+1) th terms in the expansion of $(x+1{)}^n$ are $n_{C_{r-2}},n_{C_{r-1}}$ and $n_{C_r}$ respectively.

Now $n_{C_{r-2}}:n_{C_{r-1}}:n_{C_r}=1:3:5$(Given)

$\Rightarrow \frac{n_{C_r}}{n_{C_{r-1}}}=\frac{5}{3}$ and $\frac{n_{C_{r-1}}}{n_{C_{r-2}}}=\frac{3}{1}$

$\Rightarrow \frac{n-r+1}{r}=\frac{5}{3}$ and $\frac{n-r+2}{r-1}=\frac{3}{1}$ $[\because \frac{n_{C_r}}{n_{C_{r-1}}}=\frac{n-r+1}{r}]$

$\Rightarrow 3n-8r+3=0$ and $n-4r+5=0$

Solving these for n and r, we get

n=7 and r=3.