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Given are the following geometric sequences: 13, 23.4, ...

The common ratio is 1.8, so far so good.

But how can I calculate the number of terms which are smaller then 9.6E13?

The solution says 51. I have no clue.

I'm looking for a hint to solve this. Thanks in advance.

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    Can you figure out how big the $n$th term is?2012-10-21

2 Answers 2

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Hint:

$13\cdot (1.8)^{n-1}\geq 9.6\times 10^{13}\Longrightarrow 1.8^{n-1}\geq 0.738461\times 10^{13}=:\alpha\Longrightarrow n-1\geq\frac{\log\alpha}{\log 1.8}=....$

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    I know you want to know the number of elements smaller than the given number, but knowing *the first* element bigger than it, which is the 52th, you automatically know that there are 51 elements *less* than that...so the sign is correct if one knows how to interpret it.2012-10-22
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Write it as $a_{n}=13\cdot 1.8^{n}$, $n\ge 0$. Then solve the inequality $a_{n}\ge 9.6E13$. That will tell you the first $n$ for which the terms arent smaller than 9.6E13, and it should be simple to find the answer from there

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    Indeed it does, @swisshenry2012-10-21