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Question
If Sr denotes the sum of the first r terms of an A.P. Then , S3n: (S2n − Sn) is
Options
n
3n
3
none of these
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Solution
Here, we are given an A.P. whose sum of r terms is Sr. We need to find `(S_(3n))/(S_(2n) - S_n)`.
Here we use the following formula for the sum of n terms of an A.P.
`S_n = n/2 [ 2a + (n -1 ) d]`
Where; a = first term for the given A.P.
d = common difference of the given A.P.
n = number of terms
So, first we find S3n,
`S_(3n) = (3n)/2 [ 2a + (3n - 1)d]`
`=(3n)/2 [2a + 3nd - d ]` .................(1)
Similarly,
`S_(2n) = (2n)/2 [ 2a + (2n - 1 ) d ] `
`= (2n)/2 [2a + 2nd -d]` .................(2)
Also,
`S_n = n/2 [ 2a + (n-1) d] `
`=n/2 [2a + nd - d ]`
So, using (1), (2) and (3), we get,
`(S_(3n))/(S_(2n) - S_n) = ((3n)/2 [2a + 3nd - d])/((2n)/2 [ 2a + 2nd - d ] - n/2 [ 2a + nd - d ])`
Taking `n/2` common, we get,
`(S_(3n))/(S_(2n) - S_n) =(3[2a + 3nd - d])/(2[2a + 2nd - d ]- [2a + nd - d])`
`=(3[2a + 3nd - d])/(4a + 4nd - 2d - 2a - nd + d)`
`=(3[2a + 3nd - d])/(2a + 3nd - d)`
= 3
Therefore, `(S_(3n))/(S_(2n)- S_n )= 3 `
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