Question

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

Answer 1

we know $n{C}_r=\frac{n!}{(n-r)!\left(r\right)!}$

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

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

$\frac{\frac{n!}{(n-r+1)!(r-1)!}}{\frac{n!}{(n-r)!r!}}=\frac{1}{3}$

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

$4r-n=1$

$\frac{n{C}_r}{n{C}_{r+1}}=\frac{3}{5}$

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

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

we get $8r-3n=-5$

by solving 2 equations we get $n=7,r=2$