Let-a-and-b-be-two-numbers-x-be-the-single-arithmetic-mean-of-a-and-b-Show-that-the-sum-of-n-arithmetic-means-between-a-and-b-is-nx-

Question Number 11930 by 786786AM last updated on 05/Apr/17

Let a and b be two numbers, x be the single arithmetic mean of a and b .

Show that the sum of n arithmetic means between a and b is nx .

Answered by ajfour last updated on 05/Apr/17

A_{r} = a + \frac{(b - a)}{(n + 1)} r

\underset{r = 1}{\sum^{n}} A_{r} = n a + \frac{(b - a)}{(n + 1)} . \frac{n (n + 1)}{2}

= n a + \frac{n (b - a)}{2}

= \frac{n (a + b)}{2} = n x .

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