Question

If an even number of A.M.s are inserted between two numbers whose sum is $\frac{13}{6}$, such that their sum exceeds their numbers by unity, then the number of means is

Answer 1

Let $a$ and $b$ be two numbers and $2n$ A.M.s are inserted between $a$ and $b$ then
$a,A_1,A_2,....,A_{2n},b$ be in $A.P.$
Given $a+b=\frac{13}{6}$
$A_1+A_2+....+A_{2n}=2n+1$
$\Rightarrow \frac{2n}{2}\times \frac{13}{6}=2n+1$
$\Rightarrow 13n=12n+6\Rightarrow n=6$
Hence number of means $=2n=12$