Question:
If $T_{2} / T_{3}$ in the expansion of $(a+b)^{n}$ and $T_{3} / T_{4}$ in the expansion of $(a+b)^{n+3}$ are equal, then $n=$
(a) 3
(b) 4
(c) 5
(d) 6
Solution:
(c) 5
In the expansion $(a+b)^{n}$, we have
$\frac{T_{2}}{T_{3}}=\frac{{ }^{n} C_{1} a^{n-1} \times b^{1}}{{ }^{n} C_{2} a^{n-2} \times b^{2}}$
In the expansion $(a+b)^{n+3}$, we have
$\frac{T_{3}}{T_{4}}=\frac{{ }^{n+3} C_{2} a^{n+1} b^{2}}{{ }^{n+3} C_{3} a^{n} b^{3}}$
Thus, we have
$\frac{T_{2}}{T_{3}}=\frac{T_{3}}{T_{4}}$
$\Rightarrow \frac{{ }^{n} C_{1} a}{{ }^{n} C_{2} b}=\frac{{ }^{n+3} C_{2} a}{{ }^{n+3} C_{3} b}$
$\Rightarrow \frac{2}{n-1}=\frac{3}{n+1}$
$\Rightarrow 2 n+2=3 n-3$
$\Rightarrow n=5$