# Nakafa Framework: LLM

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

Output docs content for large language models.

---

export const metadata = {
   title: "Difference between Arithmetic and Geometric Sequence",
   description: "Compare arithmetic and geometric sequences: constant differences vs ratios. Learn identification methods, formulas, and real-world applications.",
   authors: [{ name: "Nabil Akbarazzima Fatih" }],
   date: "04/05/2025",
   subject: "Sequence and Series",
};

## Understanding Sequences

A number sequence is an arrangement of numbers that follows a specific pattern. And we have already learned about two main types of sequences: [arithmetic sequences](/subject/senior-high-school/10/mathematics/sequence-series/arithmetic-sequence) and [geometric sequences](/subject/senior-high-school/10/mathematics/sequence-series/geometric-sequence).

## Arithmetic Sequences

### Definition of Arithmetic Sequences

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant.

If we have a sequence <InlineMath math="a_1, a_2, a_3, ..., a_n" />, then it is an arithmetic sequence if the difference between consecutive terms is always the same:

<BlockMath math="a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = ... = a_n - a_{n-1} = b" />

where <InlineMath math="b" /> is the constant difference.

### Formula for the nth Term of an Arithmetic Sequence

For an arithmetic sequence with first term <InlineMath math="a" /> and common difference <InlineMath math="b" />, the formula for the nth term is:

<BlockMath math="U_n = a + (n-1)b" />

## Geometric Sequences

### Definition of Geometric Sequences

A geometric sequence is a sequence of numbers where the ratio between consecutive terms is constant.

If we have a sequence <InlineMath math="a_1, a_2, a_3, ..., a_n" />, then it is a geometric sequence if the ratio between consecutive terms is always the same:

<BlockMath math="\frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = ... = \frac{a_n}{a_{n-1}} = r" />

where <InlineMath math="r" /> is the constant ratio.

### Formula for the nth Term of a Geometric Sequence

For a geometric sequence with first term <InlineMath math="a" /> and common ratio <InlineMath math="r" />, the formula for the nth term is:

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

## Key Differences

### How to Identify the Type of Sequence

To determine whether a sequence is arithmetic or geometric:

1. **Arithmetic Sequence**: Calculate the difference between consecutive terms. If the difference is always the same, then the sequence is arithmetic.

   <BlockMath math="b = a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = ..." />

2. **Geometric Sequence**: Calculate the ratio between consecutive terms. If the ratio is always the same, then the sequence is geometric.
   <BlockMath math="r = \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = ..." />

### Comparison Table

| Aspect                   | Arithmetic Sequence                    | Geometric Sequence                           |
| ------------------------ | -------------------------------------- | -------------------------------------------- |
| Pattern                  | Constant difference                    | Constant ratio                               |
| Formula for the nth term | <InlineMath math="U_n = a + (n-1)b" /> | <InlineMath math="U_n = a \times r^{n-1}" /> |
| Growth                   | Linear                                 | Exponential                                  |

## Applications in Daily Life

### Examples of Arithmetic Sequences

1. **Regular Savings**: A student saves money in the school cooperative with an arithmetic pattern. In the first month, they save Rp5,000, in the second month Rp7,000, in the third month Rp9,000, and so on. With a difference of Rp2,000, the total savings in the 10th month can be calculated using the arithmetic sequence formula.

2. **Plant Growth**: The height of a plant that increases at a constant rate each week. If a plant grows 3 cm taller every week with an initial height of 15 cm, then its height follows an arithmetic sequence.

### Examples of Geometric Sequences

1. **Investment with Compound Interest**: If Rp1,000,000 is invested with a 10% annual interest rate, the investment value will form a geometric sequence with a ratio of 1.1.

2. **Population Growth**: Bacteria that double in population every hour form a geometric sequence with a ratio of 2.
