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Math Help - Geometric progression

  1. #1
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    Geometric progression

    Question:

    A geometric progression has first term a (where a > 0), and common ratio r. The sum of the first n terms is Sn and the sum to infinity is S. Given that S2 is twice the value of the fifth term, find the value of r. Hence find the least value of n such that Sn is within 5 % of S.

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  2. #2
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    Quote Originally Posted by Tangera View Post
    Question:

    A geometric progression has first term a (where a > 0), and common ratio r. The sum of the first n terms is Sn and the sum to infinity is S. Given that S2 is twice the value of the fifth term, find the value of r. Hence find the least value of n such that Sn is within 5 % of S.

    Thank you for helping!
    You want S_2 = 2*s_5

    where S_n =\sum_0^n ar^n, and s_n = ar^n

    \sum_0^2 ar^n=2ar^5

    \sum_0^2 r^n=2r^5

    1+r+r^2=2r^5

    0=2r^5-r^2-r-1

    The r that satisfies this equation is your answer.
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  3. #3
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    Hello, Tangera!

    Are you sure of the wording of the problem?
    It doesn't seem to be solvable.


    A geometric progression has first term a > 0, and common ratio r.
    The sum of the first n terms is S_n and the sum to infinity is S.

    Given that S_2 is twice the value of the fifth term, find the value of r.
    Hence find the least value of n such that S_n is within 5% of S.

    S_2 is the sum of the first two terms: . a + ar
    The fifth term is: . a_5 \:=\:ar^4

    We have: . a + ar \:=\:2ar^4 \quad\Rightarrow\quad 2r^4 - r - 1 \:=\:0

    . . which factors: . (r-1)(2r^3 + 2r^2 + 2r + 1) \:=\:0

    We know that r \neq 1, and the cubic has no rational roots.

    Now what?

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  4. #4
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    Hello!
    Thank you for helping me!

    @ Soroban: I used the graphic calculator to solve that equation with power 4, and I could get 4 roots: 2 of which are the same (-0.17610), 1, and -0.64780...
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