Question

In the expansion of $\left(a + b\right)^{n},$ first three terms are $243,810$ and $1080$ respectively, then the fourth term of the expansion is $\left(n \in N\right)$

• A. $32$
• B. $720$
• C. $510$
• D. $420$

Answer 1

Answer: (B)

$T_{1}=a^{n},T_{2}=na^{n - 1}b,$ $T_{3}=\frac{n \left(n - 1\right)}{2}a^{n - 2}\cdot b^{2}$
Now, $\frac{T_{2}^{2}}{T_{1} T_{3}}=\frac{2 n}{n - 1}=\frac{810 \times 810}{243 \times 1080}=\frac{5}{2}$
$\Rightarrow n=5$
So, $a=3$ and $b=2$ ,
Hence, $T_{4}=\_{}^{5}C_{3}\cdot 3^{2}\cdot 2^{3}=720$