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
$\frac{{ }^{n} C_{1}}{{ }^{n} C_{2}}=\frac{{ }^{n+3} C_{2}}{{ }^{n+3} C_{3}}$
$\Rightarrow \frac{2}{n-1}=\frac{3}{n+1}$
$\Rightarrow 2 n+2=3 n-3$
$\Rightarrow n=5$
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