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21 November, 19:11

The common ratio of a geometric progression is 1/2, the fifth term is 1/80, and the sum of all of its terms is 127/320. Find the number of terms in the progression.

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Answers (2)
  1. 21 November, 19:47
    0
    This is what im getting out if it.

    1/2 * the fith term with the sum of its terms would equal your % your looking for.

    Good luck on your test!
  2. 21 November, 20:37
    0
    n = 7

    number of terms is 7

    Step-by-step explanation:

    The common ratio is 1/2 of a geometric progression. The fifth term = 1/80

    sum of all it terms = 127/320.

    The number of term in the progression can be computed as follows:

    let us get the first term

    nth term = arⁿ⁻¹

    where

    a = first term

    n = number of terms

    r = common ratio

    nth term = arⁿ⁻¹

    fifth term = 1/80

    1/80 = a * 1/2⁵⁻¹

    1/80 = a * 1/2⁴

    1/80 = a/16

    cross multiply

    16 = 80a

    divide both sides by 80

    a = 16/80

    a = 1/5

    The sum of all terms = 127/320

    sum = a (1 - rⁿ) / 1 - r

    127/320 = 1/5 (1 - 1/2ⁿ) / 1 - 1/2

    127/320 = 1/5 (1 - 1/2ⁿ) / 1/2

    127/320 = 2/5 (1 - 1/2ⁿ)

    127/320 = 2/5 - 2/5 * 1/2ⁿ

    collect like terms

    127/320 - 2/5 = - 2/5 * 1/2ⁿ

    (127 - 128) / 320 = - 2/5 * 1/2ⁿ

    -1/320 = - 2/5 * 1/2ⁿ

    multiply both sides by - 5/2

    -1/320 * - 5/2 = 1/2ⁿ

    5/640 = 1/2ⁿ

    1/128 = 1/2ⁿ

    1/2⁷ = 1/2ⁿ

    both sides have same base

    n = 7

    number of terms = 7
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