# How do you find the number of terms in the following geometric series: 100 + 99 + 98.01 + ... + 36.97?

Feb 27, 2016

Find the general form of the ${n}^{\text{th}}$ term of a geometric series and solve for $n$ when that term of $36.97$ to find that there are $100$ terms in the given series.

#### Explanation:

A geometric series is a series of the form
${a}_{0} + {a}_{0} r + {a}_{0} {r}^{2} + {a}_{0} {r}^{3} + \ldots$
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.

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

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

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

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

Thus there are $100$ terms in the series.