## Considerations of 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, then 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

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

### 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 be proving the sum of 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 \( n \ne 1 \).

## Frequently Asked Questions

### What is the sum of 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 \( 1 + 2 + 4 + 8 + \cdots + 256 \).