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URL: https://nakafa.com/en/subjects/mathematics/sequence-series/difference-sequence-series
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Clarify sequence vs series: ordered lists of terms vs their sums. Understand fundamental concepts with concrete, practical examples.

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## Sequence

**Concept:**
A sequence is simply an **ordered list of numbers** arranged according to a specific **pattern** or rule. Each number in the list is called a **term**.

Examples:

- Sequence of even numbers: $$2, 4, 6, 8, \dots$$
- Sequence of squares: $$1, 4, 9, 16, \dots$$

Visible text: - Sequence of even numbers: 
- Sequence of squares:

The focus is on **each individual term** and how the pattern forms.

Think of a sequence like **stairs**. Each step is a term, and there's a clear order from the first step, second step, and so on.

## Series

**Concept:**
With a series, we **sum up** the terms of a sequence. So, a series is the **result of the cumulative addition** of the terms in a sequence.

Examples (using the sequences above):

- Series of even numbers: $$2 + 4 + 6 + 8 + \dots$$
- Series of squares: $$1 + 4 + 9 + 16 + \dots$$

Visible text: - Series of even numbers: 
- Series of squares:

The focus is on the **total value** or **sum** of the terms up to a certain $$n$$-th term (usually denoted as $$S_n$$).

Visible text: The focus is on the **total value** or **sum** of the terms up to a certain -th term (usually denoted as ).

Back to the stairs picture. If the sequence represents the individual steps, the series represents the **total height** you've climbed after taking a certain number of steps.

## Core Difference

| Aspect      | Sequence                                                                                | Series                                                                           |
| ----------- | --------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------- |
| **Form**    | Ordered list of numbers separated by commas, such as $$U_1, U_2, U_3, \dots$$ | Sum of numbers separated by plus signs, such as $$U_1 + U_2 + U_3 + \dots$$ |
| **Focus**   | Pattern and value of each individual term                                               | Result of summing the terms                                                      |
| **How to picture it** | Individual stair steps                                                                  | Total height climbed after taking stairs                                         |

Visible text: | Aspect | Sequence | Series |
| ----------- | --------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------- |
| **Form** | Ordered list of numbers separated by commas, such as | Sum of numbers separated by plus signs, such as |
| **Focus** | Pattern and value of each individual term | Result of summing the terms |
| **How to picture it** | Individual stair steps | Total height climbed after taking stairs |