Question

Between two numbers whose sum is $\frac{13}{6}$, an even number of A.M.s is inserted, the sum of these means exceeds their number by unity. Find the number of means.

Answer 1

Let $a$ and $b$ be two numbers and $2n$ A.M.s are inserted between $a$ and $b,$ then
$\frac{2n}{2}(a+b)=2n+1.$
$n\left(\frac{13}{6}\right)=2n+1$ [given $a+b=\frac{13}{6}$]
$\Rightarrow$ $n=6$
$\therefore$ Number of means$=12.$