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Question
If S1 be the sum of (2n + 1) terms of an A.P. and S2 be the sum of its odd terms, then prove that: S1 : S2 = (2n + 1) : (n + 1).
Solution
To prove: S1 : S2 = (2n + 1) : (n + 1)
We know that the sum of AP is given by the formula:
`S = n/2(2a + (n - 1)d)`
Substituting the values in the above equation,
`S_1 = (2n + 1)/2 (2a + 2nd)`
For the sum of odd terms, it is given by,
`S_2 = a_1 + a_3 + a_5 + .....a_(2n) + 1`
`S_2 = a + a + 2d + a + 4d + .... + a + 2nd`
`S_2 = (n + 1)a + n (n + 1)d`
`S_2 = (n + 1)(a + nd)`
Hence,
`S_1 : S_2 = (2n + 1)/(n + 1)`
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