i tried every way but not able to find rigid proof ?
I will help you out by providing some notation. The arithmetic progression can be represented by the terms and the geometric progression can be represented by the terms . For a progression to be arithmetic, there must be some real number such that . For a progression to be geometric, there must be some real number with . Next, let's figure out the arithmetic and geometric sums:
I would be the first to say that I have no idea what the author of this question expects as an answer.
Because both sequences have the same first term, lets call it .
Then the last term being equal we get [TEX]a+nd-=ar^n [TEX]