anonymous
  • anonymous
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
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
oh I forgot to mention the sum to infinity of the terms of the sequence is 1000
DanJS
  • DanJS
Did you figure out the general sequence yet using those terms given
DanJS
  • DanJS
oh ok, had to look what that meant, sum to infinity is the limit of that sequence

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DanJS
  • DanJS
have you seen this before: \[S _{n} = \frac{ a _{1}(1-r^n) }{ 1-r }\] sum of n terms of geometric series
DanJS
  • DanJS
r = ratio of n+1 term to the nth term ... r = 90/100 = 81/90 = 0.9
DanJS
  • DanJS
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
DanJS
  • DanJS
sorry back,
DanJS
  • DanJS
yeah 44 looks right
DanJS
  • DanJS
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
DanJS
  • DanJS
\[990=\frac{ 100(1-0.9^n )}{ 1-0.9 }\] solve
DanJS
  • DanJS
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
DanJS
  • DanJS
i got 43.something, so 44 terms to get there
anonymous
  • anonymous
Oh I think I see. Thank you very much!
DanJS
  • DanJS
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
DanJS
  • DanJS
if r is larger than 1, then the thing will not have an infinite sum or converging value as n goes to larger
DanJS
  • DanJS
in that case it just blows up since each term is larger than the last
anonymous
  • anonymous
Ok. Good to know. :) Thanks very much!

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