A Beginner’s Guide to the Algebra of Sequences

### What is a Sequence?

A sequence is just a list of numbers in a specific order. Think of it like a list of items you write down on paper, but the items are numbers.

For example:

- Sequence A: 1, 2, 3, 4, 5, ...

- Sequence B: 2, 4, 6, 8, 10, ...

### Algebra of Sequences

The "algebra of sequences" is just a fancy way of saying **we can do basic math with these lists of numbers**. Just like you can add, subtract, multiply, or divide regular numbers, you can do the same with sequences, but you do it for each number in the list.

### Basic Operations on Sequences

#### 1. Adding Sequences

To add two sequences, you just add each number in the first list to the matching number in the second list.

- **Example**:  

  - Sequence A: 1, 2, 3, 4, 5, ...

  - Sequence B: 2, 4, 6, 8, 10, ...

  - Add them:  

    1 + 2 = 3, 2 + 4 = 6, 3 + 6 = 9, 4 + 8 = 12, 5 + 10 = 15, ...

  - New sequence: 3, 6, 9, 12, 15, ...

#### 2. Subtracting Sequences

To subtract one sequence from another, subtract each number in one list from the matching number in the other list.

- **Example**:  

  - Sequence A: 6, 7, 8, 9, 10, ...

  - Sequence B: 2, 3, 4, 5, 6, ...

  - Subtract them:  

    6 - 2 = 4, 7 - 3 = 4, 8 - 4 = 4, 9 - 5 = 4, 10 - 6 = 4, ...

  - New sequence: 4, 4, 4, 4, 4, ...

#### 3. Multiplying Sequences

To multiply two sequences, multiply each number in one list by the matching number in the other list.

- **Example**:  

  - Sequence A: 1, 2, 3, 4, 5, ...

  - Sequence B: 2, 3, 4, 5, 6, ...

  - Multiply them:  

    1 * 2 = 2, 2 * 3 = 6, 3 * 4 = 12, 4 * 5 = 20, 5 * 6 = 30, ...

  - New sequence: 2, 6, 12, 20, 30, ...

#### 4. Dividing Sequences

To divide one sequence by another, divide each number in one list by the matching number in the other list (as long as you’re not dividing by zero).

- **Example**:  

  - Sequence A: 10, 20, 30, 40, 50, ...

  - Sequence B: 2, 4, 6, 8, 10, ...

  - Divide them:  

    10 / 2 = 5, 20 / 4 = 5, 30 / 6 = 5, 40 / 8 = 5, 50 / 10 = 5, ...

  - New sequence: 5, 5, 5, 5, 5, ...

### Summary

- **Addition**: Add the numbers in the same position from each list.

- **Subtraction**: Subtract the numbers in the same position from each list.

- **Multiplication**: Multiply the numbers in the same position from each list.

- **Division**: Divide the numbers in the same position from each list.

It’s just like doing regular math, but you're working with whole lists of numbers instead of just single numbers!