Question

Prove that the difference between sum of first n terms and next n terms is n^2d

Answer 1

Let us find the pattern:

Suppose X is sum of first n terms. Let the first term be p and common difference be d.

When n=1, sum of first n terms is p; sum of first 2n terms is 2p+d; sum of first 3n terms is 3p+3d. According to our supposition, this is X, 2X+d, 3X+3d respectively.

Also, a=2X+d and b=X+2d.

When n=2, sum of first n terms is 2p+d; sum of 2n terms is4p+6d; sum of 3n terms is 6p+15d. According to our supposition, this is X, 2X+4d, 3X+12d respectively.

Also, a= 2X+4d and b= X+8d

and so on.

We are getting a pattern. a= 2X+n^2*d and b= X+2*n^2*d

Eliminating X and solving, we get d=(2*b-a)/(3*n^2)

Answer 2

i hope it is right answer