If in the expansion of
Question:

If in the expansion of (a + b)n and (a + b)n + 3, the ratio of the coefficients of second and third terms, and third and fourth terms respectively are equal, then n is

(a) 3

(b) 4

(c) 5

(d) 6

Solution:

(c) n = 5

Coefficients of the 2 nd and 3 rd terms in $(a+b)^{n}$ are ${ }^{n} C_{1}$ and ${ }^{n} C_{2}$

Coefficients of the 3 rd and 4 th terms in $(a+b)^{n+3}$ are ${ }^{n+3} C_{2}$ and ${ }^{n+3} C_{3}$

Thus, we have

$\frac{{ }^{n} C_{1}}{{ }^{n} C_{2}}=\frac{{ }^{n+3} C_{2}}{{ }^{n+3} C_{3}}$

$\Rightarrow \frac{2}{n-1}=\frac{3}{n+1}$

$\Rightarrow 2 n+2=3 n-3$

$\Rightarrow n=5$