Question

If three successive coefficients in the expansion of ${(1+x)}^n$ are $220,495$ and $792$, then find $n$.

Answer 1

Let the three successive coefficients in $(1+x{)}^n$ are ${}^n{C}_{r-1}{,}^n{C}_r{,}^n{C}_{r+1}$

$\frac{{}^n{C}_r}{{}^n{C}_{r-1}}=\frac{n-r+1}{r}=\frac{495}{220}$

$⟹\frac{n-r+1}{r}=\frac{9}{4}⟹13r-4n=4\cdots \cdots \left(1\right)$

$\frac{{}^n{C}_{r+1}}{{}^n{C}_r}=\frac{n-r}{r+1}=\frac{792}{495}$

$⟹\frac{n-r}{r+1}=\frac{8}{5}⟹5n-13r=8\cdots \cdots \left(2\right)$

$\left(1\right)+\left(2\right)⟹n=12$