If in the expansion of (1 + x)n, the coefficients of three consecutive terms are 56, 70 and 56, then find n and the position of the terms of these coefficients.
Suppose $r^{t h},(r+1)^{t h}$ and $(r+2)^{t h}$ terms are the three consecutive terms.
Their respective coefficients are ${ }^{n} C_{r-1},{ }^{n} C_{r}$ and ${ }^{n} C_{r+1}$.
We have:
${ }^{n} C_{r-1}={ }^{n} C_{r+1}=56$
$\Rightarrow r-1+r+1=n \quad\left[\right.$ If ${ }^{n} C_{r}={ }^{n} C_{s} \Rightarrow r=s$ or $\left.r+s=n\right]$
$\Rightarrow 2 r=n$
$\Rightarrow r=\frac{n}{2}$
Now,
${ }^{n} C_{\frac{n}{2}}=70$ and ${ }^{n} C_{\left(\frac{n}{2}-1\right)}=56$
$\Rightarrow \frac{{ }^{n} C_{\left(\frac{n}{2}-1\right)}}{{ }^{n} C_{\frac{n}{2}}}=\frac{56}{70}$
$\Rightarrow \frac{\frac{n}{2}}{\left(\frac{n}{2}+1\right)}=\frac{8}{10}$
$\Rightarrow 5 n=4 n+8$
$\Rightarrow n=8$
So, $r=\frac{n}{2}=4$
Thus, the required terms are 4 th, 5 th and 6 th.