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Question

Prove that the sum of an odd number of terms in A.P. is equal to the middle term multiplied by the number of terms.

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Solution

Let the AP be a,a+d,a+2d,...
The Sum of terms = n2(2a+(n1)d)
=an+n(n1)2d
Let there are n terms in the AP and n is an odd integer.
Hence, the middle term is (n+12)th term=a+(n+121)d=a+(n1)2d
Hence, multiplying n to the middle term would b equal to sum of terms.

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