Geometric Series

Concept of Geometric Series

Consider the data on the number of Covid-19 infected patients in the following table:

Month	January	February	March	April	May

The data above shows a pattern of increasing numbers of patients each month. If we sum the number of patients from the first month up to a certain month, we form a Geometric Series.

A Geometric Series is the sum of the terms of a geometric sequence. A geometric sequence is a sequence of numbers where the ratio between any two consecutive terms is always constant. This ratio is called the common ratio denoted by $(r)$ .

In the Covid-19 patient data:

The first term ( $a$ or $U_1$ ) is $4$ .
The ratio $(r) = \frac{12}{4} = \frac{36}{12} = \frac{108}{36} = \frac{324}{108} = 3$ .

So, the sequence of patient numbers is $4, 12, 36, 108, 324$ . The geometric series is the sum of the terms of this sequence:

Sum of the first $2 \text{ months}$ $(S_2) = 4 + 12 = 16$
Sum of the first $3 \text{ months}$ $(S_3) = 4 + 12 + 36 = 52$
Sum of the first $4 \text{ months}$ $(S_4) = 4 + 12 + 36 + 108 = 160$
and so on.

Finding the Formula for the Sum of the First Terms

How can we calculate the sum of the first $n$ terms $(S_n)$ without adding them one by one? Let's find the formula.

Consider this table which shows the process of rediscovering the formula for the sum of a geometric series:

Notation	Direct Summation	Using $U_{n+1}$ and $U_1$	General Form
$S_2$	$4 + 12 = 16$	$S_2 = \frac{36 - 4}{3 - 1} = \frac{32}{2} = 16$

From the bottom right of the table, we get the general form:

S_n = \frac{U_{n+1} - U_1}{r - 1}

We know that the formula for the $n$ -th term in a geometric sequence is $U_n = ar^{n-1}$ . Thus, $U_{n+1} = ar^{(n+1)-1} = ar^n$ . Substitute $U_{n+1} = ar^n$ and $U_1 = a$ into the $S_n$ formula:

S_n = \frac{ar^n - a}{r - 1}

This is the formula for the sum of the first $n$ terms of a geometric series.

Geometric Series Formula

In general, the formula to calculate the sum of the first $n$ terms of a geometric series is:

S_n = \frac{a(r^n - 1)}{r - 1}, \text{ for} r > 1

or

S_n = \frac{a(1 - r^n)}{1 - r}, \text{ for} r < 1

Legend:

$S_n$ = sum of the first $n$ terms
$a$ = first term $(U_1)$
$r$ = ratio $(r \neq 1)$
$n$ = number of terms

Example Application

A bicycle company's production in $2020$ increased monthly, forming a geometric sequence. Production in January was $120 \text{ units}$ . In April, production reached $3{,}240 \text{ units}$ . What was the total bicycle production up to May?

Solution:

January Production $(U_1) = a = 120$
April Production $(U_4) = 3{,}240$
Question: Total production up to May $(S_5)$

Step $1$ : Find the ratio $(r)$

U_4 = ar^{4-1} = ar^3

Step $2$ : Calculate $S_5$ . Since $r = 3 > 1$ , use the formula:

S_n = \frac{a(r^n - 1)}{r - 1}

Therefore, the total bicycle production up to May is $14{,}520 \text{ units}$ .