Question

The sum of two numbers is $2\frac{1}{6}$. An even number of arithmetic means are being inserted between them and their sum exceeds their number by $1$. Find the number of means inserted.

Answer 1

Let a and b be the two numbers so that $a+b=\frac{13}{6}$, given .$\left(1\right)$
Let $2n$(even) means be inserted between then whose sum $=(2n+1)$ given $\left(2\right)$
Now we know that in an A.P., sum of $2n$ means $=2n\times$ (single A.M.) by Q.$39$.
or $2n+1=2n\left(\frac{a+b}{2}\right)$ by $\left(2\right)$
or $2n+1=n\cdot \frac{13}{6}$, by $\left(1\right)$
or $12n+6=13n$,
$\therefore n=6$
Hence the number of means inserted $=2n=12$.