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Prove that the sum of n arithmetic means between two numbers is

n times the single A.M between them

1
by nishiKhan516

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n times the single A.M between them

by nishiKhan516

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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.