If d be the common difference of an A.P. and S_n be the sum of its n terms, then prove that d = S_n - 2S_n-1 + S_n-2

If the sum of first m terms of an A.P. is the same as the sum of its first n terms, show that the sum of its first (m + n) terms is zero.

In an A.P. the first term is 2, and the sum of the first five terms is one-fourth of the next five terms. Show that its 20th term is — 112.

The sum of the first 7 terms of an A.P. is 10, and that of the next 7 terms is 17. Find the progression.

If the pth term of an A.P. is x and qth term is y, show that the sum of (p + q) terms is