# 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.

96% of learners report doubling their learning speed with Shiken

• Calculus
• Geometry
• Mechanics
• Probability
• Statistics