--- title: Geometric Series --- ## 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 $\frac{a_{2}}{a_{1}} = \frac{a_{3}}{a_{2}}=...\frac{a_{n}}{a_{n-1}} = r$ $\boldsymbol{r}$ is known as common ratio of GP. if a is the first term then GP can be written as $a,ar,ar^{2},...,ar^{n-1}$ 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 $T_{n} = l = ar^{^{n-1}}$ where $r=\frac{T_{n}}{T_{n-1}}$ # 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: $S_{n}=\left\{\begin{matrix}\frac{a\left ( r^{n} -1\right )}{r-1}, r>1 & \\ \frac{a\left ( 1-r^{n} \right )}{1-r}, r<1 & \end{matrix}\right.$ and $S_{n}$ is not defined for r=1 # Geometric Mean if we insert geometric mean between two numbers $n_{1}$ and $n_{2}$ , then Geometric mean = $\sqrt{n_{1}*n_{2}}$