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.
<ahref="https://www.codecogs.com/eqnedit.php?latex=\boldsymbol{r}"target="_blank"><imgsrc="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 <ahref="https://www.codecogs.com/eqnedit.php?latex=a,ar,ar^{2},...,ar^{n-1}"target="_blank"><imgsrc="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
<ahref="https://www.codecogs.com/eqnedit.php?latex=T_{n}&space;=&space;l&space;=&space;ar^{^{n-1}}"target="_blank"><imgsrc="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 <ahref="https://www.codecogs.com/eqnedit.php?latex=r=\frac{T_{n}}{T_{n-1}}"target="_blank"><imgsrc="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:
and <ahref="https://www.codecogs.com/eqnedit.php?latex=S_{n}"target="_blank"><imgsrc="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 <ahref="https://www.codecogs.com/eqnedit.php?latex=n_{1}"target="_blank"><imgsrc="https://latex.codecogs.com/gif.latex?n_{1}"title="n_{1}"/></a> and <ahref="https://www.codecogs.com/eqnedit.php?latex=n_{2}"target="_blank"><imgsrc="https://latex.codecogs.com/gif.latex?n_{2}"title="n_{2}"/></a> , then
Geometric mean = <ahref="https://www.codecogs.com/eqnedit.php?latex=\sqrt{n_{1}*n_{2}}"target="_blank"><imgsrc="https://latex.codecogs.com/gif.latex?\sqrt{n_{1}*n_{2}}"title="\sqrt{n_{1}*n_{2}}"/></a>