# Nakafa Framework: LLM

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

Output docs content for large language models.

---

import { BacterialGrowth } from "@repo/design-system/components/contents/animation-bacterial";

export const metadata = {
  title: "Geometric Sequence",
  description: "Master geometric sequences with constant ratios. Learn formulas, solve real-world problems like bacterial growth, and discover practical applications.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/05/2025",
  subject: "Sequence and Series",
};

## Understanding Geometric Sequences

A geometric sequence is a sequence of numbers in which each term has a constant ratio to the previous term. This ratio is denoted by the letter <InlineMath math="r" />.

If we have a geometric sequence <InlineMath math="U_1, U_2, U_3, ..., U_n" />, then:

<BlockMath math="\frac{U_2}{U_1} = \frac{U_3}{U_2} = \frac{U_4}{U_3} = ... = r" />

In other words, each term in a geometric sequence is obtained by multiplying the previous term by the ratio <InlineMath math="r" />.

## Paper Folding Exploration

Let's perform a simple exploration to understand the concept of a geometric sequence. Prepare a rectangular piece of paper and fold it several times.

If the paper is folded once, it will be divided into 2 equal parts. If folded again (twice), it will form 4 equal parts. The following pattern emerges:

| Number of folds | Number of equal parts |
| --------------- | --------------------- |
| 1 fold          | 2 parts               |
| 2 folds         | 4 parts               |
| 3 folds         | 8 parts               |
| 4 folds         | 16 parts              |

Notice that the number of parts formed creates a sequence: 2, 4, 8, 16, ...

In this sequence, each term is obtained by multiplying the previous term by 2. In other words, the ratio is 2.

## General Formula for Geometric Sequences

The general formula for a geometric sequence is:

<BlockMath math="U_n = a \cdot r^{n-1}" />

Where:

- <InlineMath math="U_n" /> = nth term
- <InlineMath math="a" /> = first term
- <InlineMath math="r" /> = ratio
- <InlineMath math="n" /> = term number

## Bacterial Division

Bacteria reproduce by dividing themselves. Within two hours, one bacterial cell divides into 3 parts.

If the initial number of bacteria is 2 cells, then:

- First term (<InlineMath math="U_1" />) = 2
- Ratio (<InlineMath math="r" />) = 3

In 20 hours, division occurs 10 times (20 hours ÷ 2 hours = 10 times).

To determine the number of bacteria after 20 hours (10th term), we use the formula:

<BlockMath math="U_{10} = 2 \cdot 3^{10-1} = 2 \cdot 3^9 = 2 \cdot 19,683 = 39,366" />

So, after 20 hours, there are 39,366 bacterial cells.

<BacterialGrowth
  formulaType="geometric"
  ratio={3}
  initialCount={2}
  maxGenerations={3}
  timeInterval={2}
  labels={{
    title: "Bacterial Division",
    bacterial: "Bacteria",
    initialBacteria: "Initial Bacteria",
  }}
/>

## Properties of Geometric Sequences

### Ratio in Geometric Sequences

The ratio (<InlineMath math="r" />) in a geometric sequence is always constant and can be calculated by dividing the next term by the previous term:

<BlockMath math="r = \frac{U_2}{U_1} = \frac{U_3}{U_2} = \frac{U_4}{U_3} = ..." />

### Finding the nth Term

To find the nth term of a geometric sequence, we can use the formula:

<BlockMath math="U_n = a \cdot r^{n-1}" />

## Applications of Geometric Sequences

Geometric sequences are applied in various fields, such as:

1. Population growth (as in the bacteria example)
2. Compound interest in economics
3. Radioactive decay in physics
4. Cell growth in biology

By understanding the concept of geometric sequences, we can model and predict various phenomena involving growth or decrease with a constant ratio.

## Example Problems

### Finding the Ratio

The first term of a geometric sequence is 4 and the fourth term is 108. Determine the ratio of this sequence.

**Solution**:

Given:

- <InlineMath math="a = 4" /> (first term)
- <InlineMath math="U_4 = 108" /> (fourth term)

Using the general formula for geometric sequences:

<div className="flex flex-col gap-4">
  <BlockMath math="U_4 = a \cdot r^{4-1}" />
  <BlockMath math="108 = 4 \cdot r^3" />
  <BlockMath math="r^3 = \frac{108}{4} = 27" />
  <BlockMath math="r = \sqrt[3]{27} = 3" />
</div>

Therefore, the ratio of the geometric sequence is 3.

### Length of Rope Sections

A rope is divided into 5 parts with lengths forming a geometric sequence. If the shortest piece is 16 cm and the longest piece is 81 cm, determine the length of the third piece.

**Solution**:

Given:

- <InlineMath math="a = 16" /> (shortest piece)
- <InlineMath math="U_5 = 81" /> (longest piece)

First step, determine the ratio:

<div className="flex flex-col gap-4">
  <BlockMath math="U_5 = a \cdot r^{5-1}" />
  <BlockMath math="81 = 16 \cdot r^4" />
  <BlockMath math="r^4 = \frac{81}{16}" />
  <BlockMath math="r = \sqrt[4]{\frac{81}{16}} = \frac{3}{2}" />
</div>

Then, find the length of the third piece (<InlineMath math="U_3" />):

<div className="flex flex-col gap-4">
  <BlockMath math="U_3 = a \cdot r^{3-1}" />
  <BlockMath math="U_3 = 16 \cdot \left(\frac{3}{2}\right)^2" />
  <BlockMath math="U_3 = 16 \cdot \frac{9}{4}" />
  <BlockMath math="U_3 = 36" />
</div>

Therefore, the length of the third piece is 36 cm.
