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Arithmetic Progression 



Problem 15 of 19 


15. If S1, S2, S3 are sum of n, 2n, 3n terms of arithmetic progression series then prove that S3 = 3(S2  S1)










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Arithmetic Progression 
15. If S1, S2, S3 are sum of n, 2n, 3n terms of arithmetic progression series then prove that S3 = 3(S2  S1)
Let a be the first term and d be the common difference Now, `S_1= n/2 [ 2a + (n  1) d ]`
`S_2 = (2n)/2 [ 2a + (2n  1) d ]`
`S_3 = (3n)/2 [ 2a + (3n  1) d ]`
Now, `3(S_2  S_1)`
`= 3 [ (2n)/2 ( 2a + (2n  1) d )  n/2 ( 2a + (n  1) d ) ]`
`= 3 [ n/2 ( 2(2a)  2a ) + n/2 ( 2(2n  1) d  (n  1) d ) ]`
`= 3 [ n/2 ( 2a ) + n/2 ( 4n  2  n + 1) d ) ]`
`= 3 [ n/2 ( 2a ) + n/2 ( 3n  1) d ) ]`
`= (3n)/2 [ 2a + ( 3n  1) d ]`
`= S_3` (Proved)





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