Question

Between two numbers whose sum is $12$, an even number of A.M.s is inserted, the sum of these means exceeds their number by $20$. Find the numbers of means.

Answer 1

$x+1=12$

$2K$ is even number., $K$ is a natural number.

There are $2K$ number of $A.M.$ be $x$ & $4.$

So, $x,x+d,x+2d,\dots x+2Kd,4$ are in,

$x+(2K+1)d=y$

$x+(2K+1)d=12-x$

$2x+(2K+1)d=12.$

SUM of $A.M.:$ $x+d,x+2d,\dots x+2Kd$

$[2x+(2K+1\left)d\right]K=2K+20$

$12K=2K+20$

$10K=20$

$K=2$

$\therefore 2K=4$ are the number of means.