Question

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

Answer 1

let n a.m, between a,b are a1,a2,a3,............an
a,a1,a2,a3,.............an,b
sum of n a.m.
Sn = n[2a1 + (n-1)d]/2 ----------(1)
a.m. of a,b is (a+b)/2
b = a+(n+1)d
a.m. = [2a+(n+1)d]/2 -------------(2)
a1 = a+d put in (1)
Sn = n[2a + 2d + (n-1)d]/2
= n[2a+(n+1)d]/2 ------------(3)
from equation (2) & (3)
Sn = n*a.m.
where d = common difference