# Sequences

## Understanding Sequences

A sequence is a set of numbers that follows a specific rule and is ordered in a particular way. There are two types of sequences: arithmetic and geometric. Both types have a common factor or difference that helps determine the order of the numbers.

## Arithmetic Sequences

An arithmetic sequence has a constant difference between each number, either through addition or subtraction. For instance, 3, 7, 11, 15, 19... has a common difference of 4, while 78, 72, 66, 60, 54... has a common difference of 6. The formula to find a specific term in an arithmetic sequence is:

**nth term = first term + (n-1) * common difference**

Let's use this formula to find the 5th term in the sequence 4, 7, 10, 13, 16, 19...:

- First, identify the variables:
**n**= 5,**first term**= 4,**common difference**= 3 - Substitute the variables into the formula:
**nth term = 4 + (5-1) * 3** - Solve the equation:
**nth term = 19**

## Geometric Sequences

A geometric sequence has a constant ratio between each number, either through multiplication or division. For example, 3, 9, 27, 81, 243... has a common ratio of 3, while 39, 18, 36, 72, 144... has a common ratio of 2. The formula to find a specific term in a geometric sequence is:

**nth term = first term * (common ratio)^n-1**

To find the 4th term in the sequence 1, 2, 4, 8, 16..., we can use this formula:

- Identify the variables:
**n**= 4,**first term**= 1,**common ratio**= 2 - Substitute the variables into the formula:
**4th term = 1 * (2)^4-1 = 8**

## Recurrence Relations

A recurrence relation is used to find each term in a sequence based on a given rule and first term. It involves using previous terms to determine the next one. The formula for this is:

**nth term = f(n-1)**

For instance, if the rule for a sequence is **f(n) = 2^n + 3**, and we need to find the next five terms in the sequence, we can use the recurrence relation to calculate the terms:

**1st term = 2^0 + 3 = 4****2nd term = 2^1 + 3 = 5****3rd term = 2^2 + 3 = 7****4th term = 2^3 + 3 = 11****5th term = 2^4 + 3 = 19**

## Increasing and Decreasing Sequences

A sequence can be described as increasing if each term is greater than the previous one, and decreasing if each term is less than the previous one. For example, **7, 15, 23, 31, 39, 47** is an increasing sequence, while **15, 10, 5, 0, -5, -10** is a decreasing sequence. A sequence can also be periodic, where the terms repeat in a cycle, such as **8, 9, 10, 8, 9, 10, 8, 9, 10**.

## Real-Life Applications of Sequences

Sequences can be used to model real-life scenarios, such as savings and salaries. If the model increases by a constant amount, it will create an arithmetic sequence, and if it increases by a constant percentage, it will create a geometric sequence.

For instance, suppose a woman starts with £2000 in her savings account and adds £200 every month. In that case, we can use the formula for the nth term of an arithmetic sequence to determine how much she will have after 1 year (12 months):

- Identify the variables:
**n**= 12,**first term**= 2000,**common difference**= 200 - Substitute the variables into the formula:
**12th term = 2000 + (12-1) * 200 = 4200**

After 12 months, the woman will have £4200 in her savings account.

## Key Takeaways

- A sequence is an ordered set of numbers that follow a specific rule.
- The two main types of sequences are arithmetic and geometric.
- An arithmetic sequence increases or decreases by addition or subtraction, while a geometric sequence increases or decreases by multiplication or division.
- Formulas can be used to find a specific term within a sequence.
- Sequences can be used to model real-life scenarios.