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Q.
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
Binomial Theorem
Solution:
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 2 n < 33 $
$\Rightarrow n < 16 \cdot 5$
and $n$ is even, so $n =16$ $n =2^4$
Number of divisors $=4+1=5$