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anonymous

  • one year ago

Hello! I don't know why I am getting the wrong answer, please help! 10 .The first three terms of a geometric sequence are 100, 90 and 81. (iv) After how many terms is the sum of the sequence greater than 99% of the sum to infinity. I get -22. the answer according to the text book is 44. In a previous question it is shown that the r=9/10

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  1. anonymous
    • one year ago
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    oh I forgot to mention the sum to infinity of the terms of the sequence is 1000

  2. DanJS
    • one year ago
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    Did you figure out the general sequence yet using those terms given

  3. DanJS
    • one year ago
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    oh ok, had to look what that meant, sum to infinity is the limit of that sequence

  4. DanJS
    • one year ago
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    have you seen this before: \[S _{n} = \frac{ a _{1}(1-r^n) }{ 1-r }\] sum of n terms of geometric series

  5. DanJS
    • one year ago
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    r = ratio of n+1 term to the nth term ... r = 90/100 = 81/90 = 0.9

  6. DanJS
    • one year ago
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    since the terms are always getting smaller and smaller by that ratio,, eventually the thing will settle or 'converge' to a certain value, the farther out you go in the value of n, the closer the total will be to the limiting value of the series

  7. DanJS
    • one year ago
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    sorry back,

  8. DanJS
    • one year ago
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    yeah 44 looks right

  9. DanJS
    • one year ago
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    since they just gave you the sum to infinity is 1000, 99% of that is 990 you want that series to total 990 or more and find out how many terms it takes, n

  10. DanJS
    • one year ago
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    \[990=\frac{ 100(1-0.9^n )}{ 1-0.9 }\] solve

  11. DanJS
    • one year ago
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    you see the value for n, round up to the next whole number, that is the number of terms to get to 990, or , 99% of 1000

  12. DanJS
    • one year ago
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    i got 43.something, so 44 terms to get there

  13. anonymous
    • one year ago
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    Oh I think I see. Thank you very much!

  14. DanJS
    • one year ago
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    yes yes, if they dont give you a value for the limit or infinite sum, then you can find it by the first value in the sequence divided by (1 - r) 100/(1-0.9) = 1000

  15. DanJS
    • one year ago
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    if r is larger than 1, then the thing will not have an infinite sum or converging value as n goes to larger

  16. DanJS
    • one year ago
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    in that case it just blows up since each term is larger than the last

  17. anonymous
    • one year ago
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    Ok. Good to know. :) Thanks very much!

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