The coefficients of $r$ th, $(r+1)$ th and $(r+2 y$ th terms in the expansion of $(1+x)^n$ are ${ }^n C_{r-1},{ }^n C_r$ and ${ }^n C_{r+1}$ respectively. It is given that,
$$
\begin{aligned}
& { }^n C_{r-1}:{ }^n C_r:{ }^n C_{r+1}=2: 4: 5 \\
& \Rightarrow \quad \frac{{ }^n C_{r-1}}{{ }^n C_r}=\frac{2}{4} \\
& \Rightarrow \quad \frac{r}{n-r+1}=\frac{1}{2} \quad\left[\because \frac{{ }^n C_r}{{ }^n C_{r-1}}=\frac{n-r+1}{r}\right] \\
& \Rightarrow \quad 2 r=n-r+1 \\
& \Rightarrow \quad n=3 r-1 \\
& \text { and } \\
& \frac{{ }^n C_r}{{ }^n C_{r+1}}=\frac{4}{5} \\
& \Rightarrow \quad \frac{r+1}{n-r}=\frac{4}{5} \\
& \Rightarrow \quad 5 r+5=4 n-4 r \\
& \Rightarrow \quad 4 n=9 r+5 \\
&
\end{aligned}
$$
Solving Eqs. (i) and (ii), we get
$r=3$ and $n=8$