Question

The coefficient 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 that general term of expansion ${(a+b)}^n$ is,

$T_{r+1}{=}^n{C}_r{a}^{n-r}{b}^r$

For ${(1+x)}^n$,

$T_{r+1}{=}^n{C}_r{1}^{n-r}{x}^r$

$T_{r+1}{=}^n{C}_r{x}^r$ …… (1)

Therefore,

Coefficient of ${(r+1)}^{th}$ term ${=}^n{C}_r$

Similarly,

Coefficient of ${\left(r\right)}^{th}$ term ${=}^n{C}_{r-1}$

Coefficient of ${(r-1)}^{th}$ term ${=}^n{C}_{r-2}$

According to the question,

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

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

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

$3(r-1)=n-r+2$

$n-4r+5=0$ …… (2)

Similarly, for $r^{th}$ and ${(r+1)}^{th}$ terms,

$3n-8r+3=0$ …… (3)

Solving equations (2) and (3), we get

$r=3$ and $n=7$

Hence, the values of $n$ is $7$ and $r$ is $3$.