Prove that the sum of `n` arithmetic means between two numbers in `n` times the single. A.M. between them.

Explanation

Let ` A_(1) , A_(2) , A_(3) ,… A_(n) ` be n A.M. 's between a and b .
then ` A_(1) = a + (b-a)/(n+1)`
` A_(2) = a+ 2 ((b -a)/(n+1))`
….. ……… …… ……..
….. ……… ……. ….
`A_(n) = a+ n ((b - a)/(n+1))`
On adding , ` A_(1) + A_(2) + A_(3) + ... + A_(n) = na + (b -a) /(n +1_) (1 + 2 + 3 + ... + n)`
` = na + (b-a)/(n+1) xx (n(n+1))/(2)`
`= na + ((b - a)n)/(2) `
` = (2na + bn - an)/(2) = (n(a+ b))/(2) = n xx ` (A.M. between a and b)