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

Answer

N/a

Explanation

Let two numbers be 'a' and 'b' and 'n' A.M.'s between these numbers be `A_(1),A_(2),A_(3),…,A_(n).`
`therefore" "a,A_(1),A_(2),A_(3),...,A_(n),"be are in A.P."`
`"Now, "A_(1)+A_(2)+A_(3)+...+A_(n)`
`=(n)/(2)(A_(1)+A_(n))" (Sum of n terms of A.P.)"`
`=(n)/(2)(A_(1)-d+A_(n)+d)" (Here d = common difference"`
`=(n)/(2)(a+b)`
`nxx(a+b)/(2)`
`nxx"(A.M. of a and b)"`