Geometric Series Solutions

• A series is the sum of the terms in a sequence.

• A geometric series is the SUM of the terms in a finite geometric sequence.
• 
Remember, a finite sequence has a defined end value.
• $S_n=a_1+a_{1}r+a_1{r}^2+a_1{r}^3+ ...+a_1{r}^{n–1}$where $S_n$ is the sum of the first n-terms.

The sum of the first n-terms of a finite geometric series is: $S_n=\frac{a_1·\left(1-r{ }^n\right)}{\left(1–r\right)}$ and $S_n=\frac{a_1–r{a}_n}{\left(1–r\right)}$

In which: $a_1$ is the first term r is the common ratio $r\ne 1$

• For example, in the geometric sequence, $\left\{–80, 20, –5, 1.25\right\}$, the sum can be calculated in a few different ways because there are not many values in the sequence.
  
  Option 1: Add the terms.
  $–80+20+\left(–5\right)+1.25=–63.75$
  
  Option 2: Substitute values into either of the geometric series formulas.$S_4=\frac{a_1–\left(–{\displaystyle \frac{1}{4}}\right)a_4}{\left(1–\left(–{\displaystyle \frac{1}{4}}\right)\right)}=\frac{–80+\left({\displaystyle \frac{1}{4}}\right)1.25}{\left(1+{\displaystyle \frac{1}{4}}\right)}=–63.75$
  
  Option 3: Calculate the sum of a series.
  $S_4=–80+\left(–80\right)\left(\frac{1}{4}\right)+\left(–80\right){\left(\frac{1}{4}\right)}^2+\left(–80\right){\left(\frac{1}{4}\right)}^3+\left(–80\right){\left(\frac{1}{4}\right)}^4=–63.75$
  
  
• The sum of a geometric series can be represented using the uppercase sigma, ∑, in sigma notation.
  $$\begin{gathered} \mathrm{last} \mathrm{value} \mathrm{of} k\searrow \\ \underset{k=1}{\overset{n}{\sum }}f{\left(r\right)}^k \leftarrow \mathrm{formula} \mathrm{for} \mathrm{the} \mathrm{series} \\ \mathrm{first} \mathrm{value} \mathrm{of} k \nearrow \end{gathered}$$
• If the value of k equals a number other than one, the number of terms in the series is determined by $n–k+1$.

Example 3

Leonhard decides to start saving for a used car. He saves $75 in the first month. In subsequent months, he increases the contribution by 5% every month. Calculate the total savings after 18 months.

After 18 months, Leonhard will have a total of $2,109.93 saved.

Example 4

Example 4

Determine the specified value.  

$\sum _{k=–2}^5{\left(3\right)}^k$