# Series - Geometric series

Initial term : $a_1$

$N^{th}$ term : $a_n$

Common ratio : q

Number of terms in the series : n

Sum of thefirst n terms : $S_n$

Sum to infinity : S

1. $a_n= qa_{n-1}= a_1q^{n-1}$
2. $a_1.a_n=a_2.a_{n-1}=...=a_i.a_{n+1-i}$
3. $a_i= \sqrt {a_{i-1}.a_{i+1}}$
4. $S_n= {a_nq-a_1 \over q-1}= {a_1(q^n-1) \over q-1 }$
5. $S= \displaystyle\lim_{n \to \infty}S_n = {a_1 \over 1-q }$
For $|q|<1,$the sum S converges as $n \to \infty .$