Question

Proof that the sum of in Arithmetic means between 2given number is n time singing Arithmetic mean between them

Answer 1

Let n a.m, between a,b are a1,a2,a3,............an
Thus the series will be
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 of a.p.