If Sn denotes the sum of first n terms of an AP prove that S30 3S20 S10 Mathematics CBSE class 10 2015

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If Sn denotes the sum of first n terms of an AP, prove that S30 = 3[S20 - S10]. Mathematics CBSE class 10 2015

If Sn denotes the sum of first n terms of an AP, prove that S30 = 3[S20 - S10].

By:tarun101

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Let sum of the first n terms of an A.P, Sn = ( n / 2) [ 2a + ( n -1)d ]
Now S30 = ( 30 / 2) [ 2a + ( 30 -1)d ] ---------(1)
S20 - S10 = ( 20 / 2) [ 2a + ( 20 -1)d ] - ( 10 / 2) [ 2a + ( 10 -1)d ]
S20 - S10 = 10 [ 2a + ( 20 -1)d ] - 5 [ 2a + ( 10 -1)d ]
S20 - S10 = 10a + 190d  -  45d .
3(S20 - S10 ) = 3(10a + 145d )
3(S20 - S10 ) = 3 x 5(2a + 29d )
3(S20 - S10 ) = ( 30 / 2) [ 2a + ( 30 -1)d ] --------(2)
From (1) and (2) we get
S30 = 3(S20 - S10 )

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