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Compare arithmetic and geometric sequences: constant differences vs ratios. Learn identification methods, formulas, and real-world applications.

---

## Understanding Sequences

A number sequence is an arrangement of numbers that follows a specific pattern. Two common types are [arithmetic sequences](/en/subjects/mathematics/sequence-series/arithmetic-sequence) and [geometric sequences](/en/subjects/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 $$a_1, a_2, a_3, ..., a_n$$, then it is an arithmetic sequence if the difference between consecutive terms is always the same:

Visible text: If we have a sequence , then it is an arithmetic sequence if the difference between consecutive terms is always the same:

```math
a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = ... = a_n - a_{n-1} = b
```

where $$b$$ is the constant difference.

Visible text: where is the constant difference.

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

For an arithmetic sequence with first term $$a$$ and common difference $$b$$, the formula for the general term is:

Visible text: For an arithmetic sequence with first term and common difference , the formula for the general term is:

```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 $$a_1, a_2, a_3, ..., a_n$$, then it is a geometric sequence if the ratio between consecutive terms is always the same:

Visible text: If we have a sequence , then it is a geometric sequence if the ratio between consecutive terms is always the same:

```math
\frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = ... = \frac{a_n}{a_{n-1}} = r
```

where $$r$$ is the constant ratio.

Visible text: where is the constant ratio.

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

For a geometric sequence with first term $$a$$ and common ratio $$r$$, the formula for the general term is:

Visible text: For a geometric sequence with first term and common ratio , the formula for the general term is:

```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.

   
   
   ```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.

   
   
   ```math
   r = \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = ...
   ```

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

 
 

2. **Geometric Sequence**: Calculate the ratio between consecutive terms. If the ratio is always the same, then the sequence is geometric.

### Comparison Table

| Aspect                   | Arithmetic Sequence                    | Geometric Sequence                           |
| ------------------------ | -------------------------------------- | -------------------------------------------- |
| Pattern                  | Constant difference                    | Constant ratio                               |
| Formula for the general term | $$U_n = a + (n-1)b$$ | $$U_n = a \times r^{n-1}$$ |
| Growth                   | Linear                                 | Exponential                                  |

Visible text: | Aspect | Arithmetic Sequence | Geometric Sequence |
| ------------------------ | -------------------------------------- | -------------------------------------------- |
| Pattern | Constant difference | Constant ratio |
| Formula for the general term | | |
| 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 $$\text{Rp}5{,}000$$, in the second month $$\text{Rp}7{,}000$$, in the third month $$\text{Rp}9{,}000$$, and so on. With a difference of $$\text{Rp}2{,}000$$, the total savings in month $$10$$ 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 $$3 \text{ cm}$$ taller every week with an initial height of $$15 \text{ cm}$$, then its height follows an arithmetic sequence.

Visible text: 1. **Regular Savings**: A student saves money in the school cooperative with an arithmetic pattern. In the first month, they save , in the second month , in the third month , and so on. With a difference of , the total savings in 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 taller every week with an initial height of , then its height follows an arithmetic sequence.

### Examples of Geometric Sequences

1. **Investment with Compound Interest**: If $$\text{Rp}1{,}000{,}000$$ 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 $$2$$.

Visible text: 1. **Investment with Compound Interest**: If is invested with a annual interest rate, the investment value will form a geometric sequence with a ratio of .

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