ORIf S, denotes the sum of the first n terms of an AP prove that S30 3

1.	ORIf S, denotes the sum of the first n terms of an AP prove that S30 3 (20 S10)3 (S20 S1o)
Answer» 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)

ORIf S, denotes the sum of the first n terms of an AP prove that S30 3 (20 S10)3 (S20 S1o)

Answer»

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)