# Proof of Sum of Geometric Series by Mathematical Induction

## Considerations of the Sum of Geometric Series

The sum of geometric series is defined using $r$, the common ratio and $n$, the number of terms. The common could be any real numbers with some exceptions; the common ratio is $1$ and $0$.

If the common ratio is $1$, the series becomes the sum of constant numbers, so the series cannot be exactly referred to as a geometric series. For example, if the first term is $5$, and the common ratio is $1$, then the series becomes $5 + 5 + 5 + 5 + \cdots + 5$, so the sum of this series would be the multiply of $5$ and the number of terms. It does not need to use any specific formula to evaluate the sum.

If the common ratio is zero, the series becomes $5 + 0 + 0 + \cdots + 0$, so the sum of this series is simply $5$.
Thus our assumptions of finding the sum of geometric series are for any real number, where $r\ne 1$ and $r \ne 0$, where $r =$ the common ratio.

## Sum of Geometric Series Formula

$$a+ar+ar^2+ar^3 + \cdots + ar^n = \displaystyle \frac{a(r^{n+1}-1)}{r-1} \text{ or } \frac{a(1-r^{n+1})}{1-r}$$

## Patterns of Geometric Series

### Sum of the first two terms

\begin{align} 1-r^{2} &= (1+r)(1-r) \\ \displaystyle \therefore 1+r &= \frac{1-r^2}{1-r} \cdots (1) \end{align}

### Sum of the first three terms

\begin{align} 1-r^3 &= \left(1+r+r^2\right)(1-r) \\ \therefore 1+r+r^2 &= \displaystyle \frac{1-r^3}{1-r} \cdots (2) \end{align}

### Sum of the first four terms

\begin{align} 1-r^4 &= (1+r^2)(1-r^2) \\ &= (1+r^2)(1+r)(1-r) \\ &= (1+r + r^2 + r^3)(1-r) \\ \therefore 1+r + r^2 + r^3 &= \displaystyle \frac{1-r^4}{1-r} \cdots (3) \end{align}

### Sum of the first five terms

\begin{align} 1+r + r^2 + r^3 &= \displaystyle \frac{1-r^4}{1-r} \cdots (3) \\ (1+r + r^2 + r^3)(1-r) &= 1-r^4 \\ (1+r + r^2 + r^3)-(1+r + r^2 + r^3)r &= 1-r^4 \\ (1+r + r^2 + r^3)-\displaystyle \frac{1-r^4}{1-r} \times r &= 1-r^4 \\ (1+r + r^2 + r^3)-\frac{r-r^5}{1-r} &= 1-r^4 \\ r + r^2 + r^3 + r^4 &= \frac{r-r^5}{1-r} \\ 1 + r + r^2 + r^3 + r^4 &= 1 + \frac{r-r^5}{1-r} \\ &= \frac{1-r + r-r^5}{1-r} \\ \therefore 1 + r + r^2 + r^3 + r^4 &= \frac{1-r^5}{1-r} \cdots (4) \end{align}

### Sum of the first $n$ terms

\begin{align} 1 + r + r^2 + r^3 + \cdots + r^{n-1} &= \frac{1-r^n}{1-r} \end{align}

### Sum of the first $n+1$ terms

\begin{align} 1 + r + r^2 + r^3 + \cdots + r^{n-1} + r^n &= \frac{1- r^{n+1}}{1-r} \end{align}

## Proof of Sum of Geometric Series by Mathematical Induction

Now, we will prove the sum of the geometric series formula by mathematical induction.

$\displaystyle 1 + r + r^2 + r^3 + \cdots + r^n = \frac{1-r^{n+1}}{1-r}$

### Step 1

Show it is true for $n=1$.
\begin{align} \text{LHS} &= 1+r \\ \text{RHS} &= \displaystyle \frac{1-r^2}{1-r} \\ &= \frac{(1+r)(1-r)}{1-r} \\ &= 1+r \\ \text{LHS} &= \text{RHS} \end{align}
Therefore the formula is true for $n = 1$.

### Step 2

Assume the formula is true for $n=k$.
That is, $\displaystyle 1 + r + r^2 + r^3 + \cdots + r^k = \frac{1-r^{k+1}}{1-r}$ .

### Step 3

Show the formula is true for $n=k+1$.
That is, $\displaystyle 1 + r + r^2 + r^3 + \cdots + r^k + r^{k+1} = \frac{1-r^{k+2}}{1-r}$ .

\require{AMSsymbols} \displaystyle \begin{align} \text{LHS} &= \bbox[yellow]{1 + r + r^2 + r^3 + \cdots + r^k} + r^{k+1} \\ &= \bbox[yellow]{\frac{1-r^{k+1}}{1-r}} + r^{k+1} &\text{by the assumption} \\ &= \frac{1-r^{k+1} + r^{k+1}-r^{k+2}}{1-r} &\text{single fraction} \\ &= \frac{1-r^{k+2}}{1-r} \\ &= \text{RHS} \end{align}

Therefore the formula is true for $n= k+1$.
Hence, the formula is true for all positive integers $n \ne 1$.

### What is the sum of the geometric series formula?

The sum of the geometric series is obtained by $$a+ar+ar^2+ar^3 + \cdots + ar^n = \displaystyle \frac{a(r^{n+1}-1)}{r-1} \text{ or } \frac{a(1-r^{n+1})}{1-r}$$, where $n$ is the number of terms, $a$ is the first term, and $r$ is the common ratio of the geometric series

### What is geometric series?

Geometric series is a number sequence connected by adding subsequent terms generated by multiplying common ratio, such as $1 + 2 + 4 + 8 + \cdots + 256$. 