Question

If S1 , S2 , S3 are the sum of n terms of three AP’s, the first term of each being unity and the respective common difference being 1, 2 , 3; prove that 2 . (S1 + S 3 =2S 2(

Answer 1

Formula of summation of an A.P: $S_{n}= \frac{n}{2}[2a+(n-1)d]$,
where $S_{n}$ = Summation till n terms.
                                 a = First term of sequence
                                 d = Common Difference.

Now, using this formula, we get 
                               S1 = $\frac{n+n^{2}}{2}$,
                               S2 = $n^{2}$, &
                               S3 = $\frac{3n^{2}-n}{2}$.

Therefore, to get the desired equation, we add S1 and S3.
                    ∴ S1 + S3 = $\frac{n+n^{2}}{2} + \frac{3n^{2}-n}{2}$
                                     = $\frac{n^{2}+3n^{2}}{2}$
                                     = $\frac{4n^{2}}{2}$
                                     = $2n^{2}$
                                     = 2(S2).

∴, S1 + S3 = 2(S2).
Hence proved.
                            

Answer 2

Please see the attachment.