if there are (n – 1) terms in an ap then prove that of the sum of odd terms to the sum of even terms is (n + 1) : n^2​

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if there are (n – 1) terms in an ap then prove that of the sum of odd terms to the sum of even terms is (n + 1) : n^2​

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Mary 1 week 2021-09-13T22:41:38+00:00 1 Answer 0 views 0

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    2021-09-13T22:43:14+00:00

    There are (2n +1) terms in an arithmetic series.

    Let a , a + d , a + 2d , a + 3d , a + 4d ……. + a + (2n +1-1)d

    Here, odd terms are : a , a + 2d , a + 4d , a + 6d , …… a + 2nd

    Let number of odd terms = r and common difference = 2d

    Tr = a + (r – 1)2d

    a + 2nd = a + (r – 1)2d

    2nd/2d = r – 1

    r = n + 1

    Hence, there are (n +1) odd terms

    Now, sum of odd terms = (n+1)/2[a + a + 2nd] [∵ Sn = n/2[first term + nth term]]

    = (n + 1)(a + nd) —–(1)

    Similarly, even terms are a +d , a + 3d , a + 5d , …… a + (2n -1)d

    Number of even terms = 2n+1 – (n +1) = n

    Now, sum of even terms = n/2[a + d + a + (2n-1)d]

    = n(a + d)

    Now, sum of odd terms/sum of even terms = (n+1)(a+d)/n(a+d)

    = (n + 1)/n

    √\________Anushka❤❤

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