Question

if S1 ,S2 ,S3 are the sum of n terms of the three ap's.the first term of each being unity and the respective common difference being 1, 2, 3. prove that S1 +S3=2S2

Answer 1

Formula of summation of an A.P: ,
where  = Summation till n terms.                                 
a = First term of sequence                                 
d = Common Difference.
Now, using this formula, we get                              
 S1 = n + n^2 / 2                               
S2 = n^2                               
S3 = 3n^2 - n /2
Therefore, to get the desired equation, we add S1 and S3.                   
∴ S1 + S3 = n + n^2 / 2 + 3n^2 - n /2                                     
= n^2 + 3n^2                                     
= 4n^2 / 2                                   
 =  2n^2                                   
 = 2(S2).
∴ S1 + S3 = 2(S2).Hence proved.