arithmetic

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there are n arithmetic means between 5 and 32 such that the ratio between the 3rd and 7th means is 7:13,find the value of n.

  • arithmetic -

    the sequence would be

    5 , 5+d , 5+2d , 5+3d , .... , 5 + (n-2)d , 32
    for n+2 terms

    3rd mean = 4th term = a+3d
    7th mean = 8th term = a + 7d
    but a = 5

    (5+3d)/(5+7d) = 7/13
    35 + 49d = 65 +39d
    10d = 30
    d = 30/10 = 3

    so a + ((n+2)-1)d = 32
    5 + (n+1)(3) = 32
    3n+3 = 27
    3n = 24
    n = 8

    check:

    5 8 11 14 17 20 23 26 29 32

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