Question

If $(x+a{)}^n$ has only one middle term and only seventh term is numerically greatest term when $x=3$, $a=2$, then number of positive divisors of $n$ is

• A. 4
• B. 5
• C. 6
• D. 7

Answer 1

Answer: (B)

One middle term $\Rightarrow n=$ even
$x=3;a=2;7^{\text{th }}$ term
${}^n{C}_5\cdot 3^{n-5}\cdot 2^5<{}^n{C}_6\cdot 3^{n-6}\cdot 2^6>{}^n{C}_7\cdot 3^{n-7}\cdot 2^7$
$\Rightarrow \frac{3}{2}<\frac{n!}{6!(n-6)!}\times \frac{5!(n-5)!}{n!}$
$\Rightarrow 2(n-5)>3\cdot 6$
$\Rightarrow n-5>9$
$\Rightarrow n>14$
$\frac{3}{2}>\frac{n!}{7!(n-7)!}\times \frac{6!(n-6)!}{n!}$
$\Rightarrow 2(n-6)<21$
$\Rightarrow 2n<33$
$\Rightarrow n<16\cdot 5$
and $n$ is even, so $n=16$ $n=2^4$
Number of divisors $=4+1=5$