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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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 $r=\frac{99}{100}=0.99$

Now, looking at the general form of the series, we can see that the $n}^{\text{th}$ 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.

$36.97={a}_{0}{r}^{n-1}=100{\left(0.99\right)}^{n-1}$

$\Rightarrow 0.3697={0.99}^{n-1}$

$\Rightarrow \mathrm{ln}\left(0.3697\right)=\mathrm{ln}\left({0.99}^{n-1}\right)=(n-1)\mathrm{ln}\left(0.99\right)$

$\Rightarrow n=\frac{\mathrm{ln}\left(0.3697\right)}{\mathrm{ln}\left(0.99\right)}+1\approx 100$

Thus there are 100 terms in the series.

${a}_{0}+{a}_{0}r+{a}_{0}{r}^{2}+{a}_{0}{r}^{3}+...$

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 $r=\frac{99}{100}=0.99$

Now, looking at the general form of the series, we can see that the $n}^{\text{th}$ 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.

$36.97={a}_{0}{r}^{n-1}=100{\left(0.99\right)}^{n-1}$

$\Rightarrow 0.3697={0.99}^{n-1}$

$\Rightarrow \mathrm{ln}\left(0.3697\right)=\mathrm{ln}\left({0.99}^{n-1}\right)=(n-1)\mathrm{ln}\left(0.99\right)$

$\Rightarrow n=\frac{\mathrm{ln}\left(0.3697\right)}{\mathrm{ln}\left(0.99\right)}+1\approx 100$

Thus there are 100 terms in the series.

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