Q18 of 110 Page 53

If there are (2n + 1) terms in an arithmetic series, then prove that the ratio of the sum of odd terms to the sum of even terms is (n + 1) : n.


In the A.P, let

First term = a


Common difference = d


Number of terms = (2n + 1)


Series: a,a + d,a + 2d……a + 2nd


For Odd terms


: a, a + 2d,…a + 2nd


First term = a


Common difference = 2d


Number of terms = n + 1


Sum of terms =


Sum of odd terms =


⇒ Sum of odd terms =


For Even terms


: a + d, a + 3d,…a + (2n–1)d


First term = a + d


Common difference = 2d


Number of terms = n


Sum of terms =


Sum of even terms =


⇒ Sum of even terms =


Sum of odd terms : Sum of even terms = : =


∴ Sum of odd terms : Sum of even terms = (n + 1) : n


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