Geometric Sequences

Let $T(1)$, $T(2)$, …, $T(n)$ be an geometric sequence and $r$ be the common ratio. For natural number $n$ and $n>1$,

1. The general term
   \begin{equation*}
   T(n)=T(1)r^{n-1}
   \end{equation*}

2. The sum of the first $n$ term
   \begin{equation*}
   S(n)=\frac{T(1)(1-r^n)}{1-r}
   \end{equation*}
   or
   \begin{equation*}
   S(n)=\frac{T(1)(r^n-1)}{r-1}
   \end{equation*}

3. The sum to infinity
   \begin{equation*}
   S(\infty) =\frac{T(1)}{1-r}\ \mbox{ for } -1 < r < 1 \end{equation*}

Same Topic:

Arithmetic Sequences Fibonacci Sequence Square Numbers Triangle Numbers