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Let a1 , a2 , ........... , an , an+1 , ....... be an A.P. Let  S1 = a1 + a2 + a3 + ............. + an  S2 = an+1 + an+2 + ...............+ a2n  S3 = a2n+1 + a2n+2 + .............+ a3n ................................................ ................................................ Prove that the sequence S1 , S2 , S3 , ........ is an arithmetic progression whose common difference is n2 times the common difference of the given progression
Asked by rushabhjain.a | 20 Feb, 2019, 09:07: PM
answered-by-expert Expert Answer
S1 = (n/2)[2 a1 +(n-1)d ]
 
S2 = (n/2) [ 2 an+1 +(n-1)d ]
 
S3 = (n/2) [ 2 a2n+1 + (n-1)d ]
 
S2 - S1 = n [an+1 - a1 ] = n [ a1 + nd - a1 ] = n2d
 
S3 - S2 = n [ a2n+1 - an+1 ] = n [ a1 + 2n d - a1 - nd ] = n2d
 
Hence it is proved that S1 , S2 and S3 are in A.P with common difference n2d
Answered by Thiyagarajan K | 21 Feb, 2019, 10:41: AM

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