Question

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

Answer 1

Let the middle term be a and common difference be d

The term before it will be a-d,a-2d,a-3d......a-nd

The terms after the middle term are a+d,a+2d,a+3d...a+nd

So the A.P is:

a-nd,a-(n-1)d,.....a-3d,a-2d,a-d,a,a+d,a+2d,a+3d...a+(n-1)d,a+nd

sum=a-nd+a-(n-1)d+.....a-3d+a-2d+a-d+a+a+d+a+2d+a+3d...a+(n-1)d+a+nd

Note:For every +d there is -d ,for every +2d there is -2d.....

So the d's gets cut and what remains is as follows:

sum=a+a+a+a+a....a+a+a+a+a=a x no.of terms

Thus, the sum of an odd number of terms in AP is equal to the middle term multiplied by the number of terms.

Hope it helps:)