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 $\left(even\right)$ means be inserted between them whose sum
$=(2n+1)$ given $...\left(2\right)$
Now we know that in an A.P., sum of 2n
means $=2n\times (singleA.M,)$
or $2n+1=2n\left(\frac{a+b}{2}\right)$ by
$\left(2\right)$ or $2n+1=n.\frac{13}{6},$ by $\left(1\right)$ or
$12n+6=13n,\therefore n=6.$
Hence the number of means
inserted $=2n=12.$