Question

If the three consecutive coefficient in the expansion of $(1+x{)}^n$ are $28,56$ and $70,$ then the value of $n$ is

Answer 1

Let the three consecutive coefficient be
${}^n{C}_{r-1}=28,{}^n{C}_r=56$ and ${}^n{C}_{r+1}=70$
$\therefore \frac{{}^n{C}_r}{{}^n{C}_{r-1}}=\frac{n-r+1}{r}=\frac{56}{28}=2\text{and}\frac{{}^n{C}_{r+1}}{{}^n{C}_r}=\frac{n-r}{r+1}=\frac{70}{56}=\frac{5}{4}$
This gives $n+1=3r$ and $4n-5=9r$
$\therefore \frac{4n-5}{n+1}=3\Rightarrow n=8$