freeCodeCamp/guide/english/mathematics/geometric-series/index.md

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title: Geometric Series
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## Geometric Series


A geometric progression is a sequence of numbers, whose first term is non zero and each term is obtained by multiplying its proceding term by a constant quantity. This constant quantity is called common ratio of the GP.

The sum of this progression is called geometric series.

We can say if a1,a2,a3,....,an are in GP then 

<a href="https://www.codecogs.com/eqnedit.php?latex=\frac{a_{2}}{a_{1}}&space;=&space;\frac{a_{3}}{a_{2}}=...\frac{a_{n}}{a_{n-1}}&space;=&space;r" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\frac{a_{2}}{a_{1}}&space;=&space;\frac{a_{3}}{a_{2}}=...\frac{a_{n}}{a_{n-1}}&space;=&space;r" title="\frac{a_{2}}{a_{1}} = \frac{a_{3}}{a_{2}}=...\frac{a_{n}}{a_{n-1}} = r" /></a>

<a href="https://www.codecogs.com/eqnedit.php?latex=\boldsymbol{r}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\boldsymbol{r}" title="\boldsymbol{r}" /></a> is known as common ratio of GP.
 
 if a is the first term then GP can be written as <a href="https://www.codecogs.com/eqnedit.php?latex=a,ar,ar^{2},...,ar^{n-1}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?a,ar,ar^{2},...,ar^{n-1}" title="a,ar,ar^{2},...,ar^{n-1}" /></a>

example of a GP : 2,4,8,16,....

## The nth term of GP (Also known as General term)

Let a be the first term, r be the common ratio and l be the last term of a GP, then nth term is given by

 <a href="https://www.codecogs.com/eqnedit.php?latex=T_{n}&space;=&space;l&space;=&space;ar^{^{n-1}}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?T_{n}&space;=&space;l&space;=&space;ar^{^{n-1}}" title="T_{n} = l = ar^{^{n-1}}" /></a>
 
 where <a href="https://www.codecogs.com/eqnedit.php?latex=r=\frac{T_{n}}{T_{n-1}}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?r=\frac{T_{n}}{T_{n-1}}" title="r=\frac{T_{n}}{T_{n-1}}" /></a>
 
 # The sum of n terms of a GP
 
 Let a be the first term, r be the common ratio and l be the last term of a GP, then sum of n terms is given by:
 <a href="https://www.codecogs.com/eqnedit.php?latex=S_{n}=\left\{\begin{matrix}\frac{a\left&space;(&space;r^{n}&space;-1\right&space;)}{r-1},&space;r>1&space;&&space;\\&space;\frac{a\left&space;(&space;1-r^{n}&space;\right&space;)}{1-r},&space;r<1&space;&&space;\end{matrix}\right." target="_blank"><img src="https://latex.codecogs.com/gif.latex?S_{n}=\left\{\begin{matrix}\frac{a\left&space;(&space;r^{n}&space;-1\right&space;)}{r-1},&space;r>1&space;&&space;\\&space;\frac{a\left&space;(&space;1-r^{n}&space;\right&space;)}{1-r},&space;r<1&space;&&space;\end{matrix}\right." title="S_{n}=\left\{\begin{matrix}\frac{a\left ( r^{n} -1\right )}{r-1}, r>1 & \\ \frac{a\left ( 1-r^{n} \right )}{1-r}, r<1 & \end{matrix}\right." /></a>
 
 and <a href="https://www.codecogs.com/eqnedit.php?latex=S_{n}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?S_{n}" title="S_{n}" /></a> is not defined for r=1

# Geometric Mean
 
if we insert geometric mean between two numbers  <a href="https://www.codecogs.com/eqnedit.php?latex=n_{1}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?n_{1}" title="n_{1}" /></a> and <a href="https://www.codecogs.com/eqnedit.php?latex=n_{2}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?n_{2}" title="n_{2}" /></a> , then 

Geometric mean = <a href="https://www.codecogs.com/eqnedit.php?latex=\sqrt{n_{1}*n_{2}}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\sqrt{n_{1}*n_{2}}" title="\sqrt{n_{1}*n_{2}}" /></a>