Question

The sum of first n terms of a sequence is 5(3^n - 1) show that the sequence is G.P

Answer 1

Answer:

The method indeed appears to assume what is to be proved. I don't see an easy way to explain away that problem in the equations as presented.

On the other hand, consider your procedure as a method for guessing what the geometric progression might be. You obtained the guess arn where

ra==13,3(1−r)=2.

Now compute the sum of the first n terms of the geometric progression 2(13)n. Is that sum 3(1−3−n)? If so, then 2(13)n is the correct sequence, for this reason: if Sn−1 and Sn are respectively the sums of the first n−1 and the first n terms of a sequence, respectively, then the nth term of that sequence is Sn−Sn−1; that is, the series of sums of a sequence uniquely determines the sequence; and the formula 3(1−3−n) describes the complete series of sums of the desired sequence.

This still seems a bit clumsy when you have the alternative to compute Sn−Sn−1 directly, as in the answer by Vikram.