# Nakafa Framework: LLM

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

Output docs content for large language models.

---

import TableChairsAnimation from "./animation-table";

export const metadata = {
  title: "Sequence Concept",
  description: "Explore sequence fundamentals with interactive table-chair patterns. Learn arithmetic, geometric types, and solve real-world pattern problems step-by-step.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/05/2025",
  subject: "Sequence and Series",
};

## Understanding Number Sequences

A number sequence is an arrangement of numbers that follows a specific pattern. Each number in the sequence is called a term. Note the following notations:

- The 1st term is denoted by <InlineMath math="U_1" />
- The 2nd term is denoted by <InlineMath math="U_2" />
- The 3rd term is denoted by <InlineMath math="U_3" />
- The nth term is denoted by <InlineMath math="U_n" />

By understanding the pattern in a sequence, we can determine the subsequent terms and even find the nth term using a general formula.

## Number Patterns in Everyday Life

### Tables and Chairs Exploration

Let's observe an example of a number pattern formed by the arrangement of tables and chairs:

<TableChairsAnimation
  labels={{
    title: "Table and Chair Sequence Pattern",
    table: "Table",
    chair: "Chair",
  }}
/>

When there is 1 square table, 4 chairs can be placed around it.

If 2 tables are joined together, then 6 chairs can be placed around the combined tables.

We can create a table to observe the pattern:

| Number of tables | 1   | 2   | 3   | 4   | 5   | 6   |
| ---------------- | --- | --- | --- | --- | --- | --- |
| Number of chairs | 4   | 6   | 8   | 10  | 12  | 14  |

From the table above, we can observe that:

- When there is 1 table, there are 4 chairs
- When there are 2 tables, there are 6 chairs
- When there are 3 tables, there are 8 chairs

When observed, each addition of 1 table results in an addition of 2 chairs. This forms a number pattern with the formula:

<BlockMath math="U_n = 2n + 2" />

Where:

- <InlineMath math="U_n" /> is the number of chairs
- <InlineMath math="n" /> is the number of tables

## Applications of Sequence Concepts

### Determining the Number of Chairs and Tables

By understanding sequence patterns, we can answer questions such as:

1. If there are 20 people who will sit on chairs, how many tables need to be joined?

We can use the formula <InlineMath math="U_n = 2n + 2" /> where <InlineMath math="U_n = 20" />, so:

<div className="flex flex-col gap-4">
  <BlockMath math="20 = 2n + 2" />
  <BlockMath math="18 = 2n" />
  <BlockMath math="n = 9" />
</div>

Therefore, 9 tables need to be joined to accommodate 20 people.

## Types of Sequences

Based on their patterns, number sequences can be classified into several types:

### Arithmetic Sequence

A number sequence where the difference between two consecutive terms is always constant. This difference is called the common difference (<InlineMath math="b" />).

Example: <InlineMath math="2, 4, 6, 8, 10, \ldots" /> Common difference <InlineMath math="(b) = 2" />

### Geometric Sequence

A number sequence where the ratio between two consecutive terms is always constant. This ratio is called the common ratio (<InlineMath math="r" />).

Example: <InlineMath math="2, 6, 18, 54, \ldots" /> Common ratio <InlineMath math="(r) = 3" />

### Other Sequences

Besides arithmetic and geometric sequences, there are many other types of sequences such as Fibonacci sequences, quadratic sequences, cubic sequences, and more.

Example of a Fibonacci sequence: <InlineMath math="0, 1, 1, 2, 3, 5, 8, 13, \ldots" />

## Finding Patterns

To determine the pattern of a sequence:

1. Observe the differences between consecutive terms
2. Check if the difference is constant (arithmetic sequence)
3. If not, check if the ratio is constant (geometric sequence)
4. If neither, check for other possible patterns

By understanding sequence concepts, we can solve various mathematical problems related to number patterns in everyday life.