7. How many arithmetic means should be insterted between 5 and 83, so that the ratio between the first and the last arithmeti

Question

7.
How many arithmetic means should be
insterted between 5 and 83, so that the ratio
between the first and the last arithmetic mean
be 1: 7.​

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Aaliyah 1 month 2021-08-12T05:51:28+00:00 1 Answer 0 views 0

Answers ( )

    0
    2021-08-12T05:53:26+00:00

    Question : How many arithmetic means should be insterted between 5 and 83, so that the ratio between the first and the last arithmetic mean be 1: 7. ?

    Solution :

    given that,

    • first term = a = 5
    • last term = an = 83 .
    • Let common difference = d .
    • second term : second last term = 1 : 7 .

    so,

    → second term = first term + common difference = (5 + d)

    → second last term = last term – common difference = (83 – d)

    A/q,

    → (5 + d) / (83 – d) = 1/7

    → 7(5 + d) = 83 – d

    → 35 + 7d = 83 – d

    → 7d + d = 83 – 35

    → 8d = 48

    → d = 6 .

    therefore,

    → Last term = an = 83

    → an = a + (n – 1)d

    → 5 + (n – 1)6 = 83

    → 5 + 6n – 6 = 83

    → 6n – 1 = 83

    → 6n = 83 + 1

    → 6n = 84

    → n = 14 .

    hence,

    → Number of arithmetic means inserted between 5 and 84 = 14 – 2 = 12 (Ans.)

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