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URL: https://nakafa.com/en/subjects/mathematics/sequence-series/geometric-series
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Learn geometric series formulas and calculate sums efficiently. Solve production problems, understand COVID-19 growth patterns, and learn applications.

---

## 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 |
| :----------------- | :------ | :------- | :---- | :---- | :-- |
| Number of patients | $$4$$ | $$12$$ | $$36$$ | $$108$$ | $$324$$ |

Visible text: | Month | January | February | March | April | May |
| :----------------- | :------ | :------- | :---- | :---- | :-- |
| Number of patients | | | | | |

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 $$(r)$$.

Visible text: 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 .

In the Covid-19 patient data:

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

Visible text: - The first term ( or ) is .
- The ratio .

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

Visible text: So, the sequence of patient numbers is .
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.

Visible text: - Sum of the first 
- Sum of the first 
- Sum of the first 
- 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.

Visible text: How can we calculate the sum of the first terms 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                                            |
| :--------------------------- | :-------------------------------------------- | :---------------------------------------------------------------------- | :------------------------------------------------------ |
| $$S_2$$    | $$4 + 12 = 16$$             | $$S_2 = \frac{36 - 4}{3 - 1} = \frac{32}{2} = 16$$    | $$S_2 = \frac{U_3 - U_1}{r - 1}$$     |
| $$S_3$$    | $$4 + 12 + 36 = 52$$        | $$S_3 = \frac{108 - 4}{3 - 1} = \frac{104}{2} = 52$$  | $$S_3 = \frac{U_4 - U_1}{r - 1}$$     |
| $$S_4$$    | $$4 + 12 + 36 + 108 = 160$$ | $$S_4 = \frac{324 - 4}{3 - 1} = \frac{320}{2} = 160$$ | $$S_4 = \frac{U_5 - U_1}{r - 1}$$     |
| $$\vdots$$ | $$\dots$$                   | $$\dots$$                                             | $$\dots$$                             |
| $$S_n$$    | $$\dots$$                   | $$\dots$$                                             | $$S_n = \frac{U_{n+1} - U_1}{r - 1}$$ |

Visible text: | Notation | Direct Summation | Using and | General Form |
| :--------------------------- | :-------------------------------------------- | :---------------------------------------------------------------------- | :------------------------------------------------------ |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |

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

```math
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:

Visible text: We know that the formula for the -th term in a geometric sequence is . Thus, .
Substitute and into the formula:

Component: MathContainer
Children:

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

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

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

Visible text: This is the formula for the sum of the first 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:

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

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

or

```math
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

Visible text: - = sum of the first terms
- = first term 
- = ratio 
- = 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?

Visible text: A bicycle company's production in increased monthly, forming a geometric sequence. Production in January was . In April, production reached . 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 $$(S_5)$$

Visible text: - January Production 
- April Production 
- Question: Total production up to May

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

Visible text: **Step** : Find the ratio

Component: MathContainer
Children:

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

```math
3.240 = 120 \cdot r^3
```

```math
r^3 = \frac{3.240}{120}
```

```math
r^3 = 27
```

```math
r = \sqrt[3]{27} = 3
```

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

Visible text: **Step** : Calculate . Since , use the formula:

Component: MathContainer
Children:

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

```math
S_5 = \frac{120(3^5 - 1)}{3 - 1}
```

```math
S_5 = \frac{120(243 - 1)}{2}
```

```math
S_5 = \frac{120(242)}{2}
```

```math
S_5 = 60 \cdot 242
```

```math
S_5 = 14.520
```

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

Visible text: Therefore, the total bicycle production up to May is .