Insert n A.M.'s between `n^2 and 1`

Insert 3 A.M.’s between 3 and 19.

Insert 10 A.M.’s between 5 and -17 and prove that their sum is ten times the A.M. between them.

There are n AM's between 3 and 54.Such that the 8th mean and (n-2) th mean is 3 ratio 5. Find n.

Find the A.M. between : 6 and -8.

If A_1 and A_2 are two A.M.’s between a and b, prove that : A_1+A_2 =a+b .

If A_1 and A_2 are two A.M.’s between a and b, prove that : (2A_1-A_2)(2A_2-A_1) =ab .

If A is the A.M. between a and b, prove that : (A-a)^(2) +(A-b)^(2)=1/2(a-b)^(2) .

If A is the A.M. between a and b, prove that : 4(a-A)(A-b)=(a-b)^(2) .

a, b, c are in G.P. and x and y are the A.M.’s between a, b and b, c respectively. Show that : a/x+c/y=2 .

a, b, c are in G.P. and x and y are the A.M.’s between a, b and b, c respectively. Show that : 1/x+1/y=2/b .