Find the number of terms in the following geometric series: 100 + 99 + 98.01 + ... + 36.97

4enevi 2022-10-17 Answered
Find the number of terms in the following geometric series: 100 + 99 + 98.01 + ... + 36.97
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Answers (1)

exalantaswo
Answered 2022-10-18 Author has 14 answers
A geometric series is a series of the form
a 0 + a 0 r + a 0 r 2 + a 0 r 3 + ...
where a 0 is the initial term and r is the common ratio between terms. We can easily find r by dividing any term after the first by the prior term. So in this case we have
a 0 = 100 and r = 99 100 = 0.99
Now, looking at the general form of the series, we can see that the n th term has the form a 0 r n - 1 . Thus, as we have the last term in the series, we simply need to solve for n for that term to find the total number of terms.
36.97 = a 0 r n - 1 = 100 ( 0.99 ) n - 1
0.3697 = 0.99 n - 1
ln ( 0.3697 ) = ln ( 0.99 n - 1 ) = ( n - 1 ) ln ( 0.99 )
n = ln ( 0.3697 ) ln ( 0.99 ) + 1 100
Thus there are 100 terms in the series.
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