# Nakafa Framework: LLM

URL: /en/subject/high-school/10/mathematics/sequence-series/geometric-series
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/sequence-series/geometric-series/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Geometric Series",
  description: "Learn geometric series formulas and calculate sums efficiently. Solve production problems, understand COVID-19 growth patterns, and master applications.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/08/2025",
  subject: "Sequence and 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 |
| :----------------- | :------ | :------- | :---- | :---- | :-- |
| Number of patients | 4       | 12       | 36    | 108   | 324 |

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 <InlineMath math="(r)" />.

In the Covid-19 patient data:

- The first term (<InlineMath math="a" /> or <InlineMath math="U_1" />) is 4.
- The ratio <InlineMath math="(r)" /> = <InlineMath math="\frac{12}{4} = \frac{36}{12} = \frac{108}{36} = \frac{324}{108} = 3" />.

So, the sequence of patient numbers is <InlineMath math="4, 12, 36, 108, 324" />.
The geometric series is the sum of the terms of this sequence:

- Sum of the first 2 months <InlineMath math="(S_2)" /> = <InlineMath math="4 + 12 = 16" />
- Sum of the first 3 months <InlineMath math="(S_3)" /> = <InlineMath math="4 + 12 + 36 = 52" />
- Sum of the first 4 months <InlineMath math="(S_4)" /> = <InlineMath math="4 + 12 + 36 + 108 = 160" />
- and so on.

## Finding the Formula for the Sum of the First n Terms

How can we calculate the sum of the first <InlineMath math="n" /> terms <InlineMath math="(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 <InlineMath math="U_{n+1}" /> and <InlineMath math="U_1" />       | General Form                                            |
| :--------------------------- | :-------------------------------------------- | :---------------------------------------------------------------------- | :------------------------------------------------------ |
| <InlineMath math="S_2" />    | <InlineMath math="4 + 12 = 16" />             | <InlineMath math="S_2 = \frac{36 - 4}{3 - 1} = \frac{32}{2} = 16" />    | <InlineMath math="S_2 = \frac{U_3 - U_1}{r - 1}" />     |
| <InlineMath math="S_3" />    | <InlineMath math="4 + 12 + 36 = 52" />        | <InlineMath math="S_3 = \frac{108 - 4}{3 - 1} = \frac{104}{2} = 52" />  | <InlineMath math="S_3 = \frac{U_4 - U_1}{r - 1}" />     |
| <InlineMath math="S_4" />    | <InlineMath math="4 + 12 + 36 + 108 = 160" /> | <InlineMath math="S_4 = \frac{324 - 4}{3 - 1} = \frac{320}{2} = 160" /> | <InlineMath math="S_4 = \frac{U_5 - U_1}{r - 1}" />     |
| <InlineMath math="\vdots" /> | <InlineMath math="\dots" />                   | <InlineMath math="\dots" />                                             | <InlineMath math="\dots" />                             |
| <InlineMath math="S_n" />    | <InlineMath math="\dots" />                   | <InlineMath math="\dots" />                                             | <InlineMath math="S_n = \frac{U_{n+1} - U_1}{r - 1}" /> |

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

<BlockMath math="S_n = \frac{U_{n+1} - U_1}{r - 1}" />

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

<div className="flex flex-col gap-4">
  <BlockMath math="S_n = \frac{ar^n - a}{r - 1}" />
  <BlockMath math="S_n = \frac{a(r^n - 1)}{r - 1}" />
</div>

This is the formula for the sum of the first <InlineMath math="n" /> terms of a geometric series.

## Geometric Series Formula

In general, the formula to calculate the sum of the first <InlineMath math="n" /> terms of a geometric series is:

<BlockMath math="S_n = \frac{a(r^n - 1)}{r - 1}, \text{ for } r > 1" />

or

<BlockMath math="S_n = \frac{a(1 - r^n)}{1 - r}, \text{ for } r < 1" />

**Legend:**

- <InlineMath math="S_n" /> = sum of the first <InlineMath math="n" /> terms
- <InlineMath math="a" /> = first term <InlineMath math="(U_1)" />
- <InlineMath math="r" /> = ratio <InlineMath math="(r \neq 1)" />
- <InlineMath math="n" /> = number of terms

## Example Application

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

**Solution:**

- January Production <InlineMath math="(U_1)" /> = <InlineMath math="a = 120" />
- April Production <InlineMath math="(U_4)" /> = 3,240
- Question: Total production up to May <InlineMath math="(S_5)" />

**Step 1**: Find the ratio <InlineMath math="(r)" />

<div className="flex flex-col gap-4">
  <BlockMath math="U_4 = ar^{4-1} = ar^3" />
  <BlockMath math="3.240 = 120 \cdot r^3" />
  <BlockMath math="r^3 = \frac{3.240}{120}" />
  <BlockMath math="r^3 = 27" />
  <BlockMath math="r = \sqrt[3]{27} = 3" />
</div>

**Step 2**: Calculate <InlineMath math="S_5" />. Since <InlineMath math="r = 3 > 1" />, use the formula:

<div className="flex flex-col gap-4">
  <BlockMath math="S_n = \frac{a(r^n - 1)}{r - 1}" />
  <BlockMath math="S_5 = \frac{120(3^5 - 1)}{3 - 1}" />
  <BlockMath math="S_5 = \frac{120(243 - 1)}{2}" />
  <BlockMath math="S_5 = \frac{120(242)}{2}" />
  <BlockMath math="S_5 = 60 \cdot 242" />
  <BlockMath math="S_5 = 14.520" />
</div>

Therefore, the total bicycle production up to May is 14,520 units.
