Sir\Madam please explain me this question on Arithmetic Progressions!

let the first term and common diff be a and d respectively.

let the number of terms be odd, say, 2n+1

then the n+1 th term is the middle term.=a+[(n+1)-1]=a+n

so we have to show that

 sum of all the 2n+1 terms =the n+1th term (2n+1)

LHS=[(2n+1)/2][a+{a+2n}]....using the formula for sum of n terms of an A.P.

=(2n+1)(a+n)

=number of terms (the middle term)