Sequences

A sequence is an ordered list of numbers. Think of it as a line of students queuing up for lunch in the school canteen; the first one in line is the first term, the second one is the second term, and so on.

In maths, we use sequences to understand patterns and structures in numbers.

Types of Sequences

At this stage, it’s good to know that there are different types of sequences you might encounter. Two common types are

• Arithmetic sequences

• Geometric sequences

In arithmetic sequences, each term after the first is found by adding a fixed amount to the previous term. In geometric sequences, each term after the first is found by multiplying the previous term by a fixed amount.

Terms in a Sequence

In any sequence, individual numbers are known as terms. So, if we have a sequence like 2, 4, 6, 8, then 2 is the first term, 4 is the second term, 6 is the third term, and so on.

The position of each number in the list is important because sequences are all about order.

Common Differences and Ratios

You’ll often hear about “common differences” in arithmetic sequences and “common ratios” in geometric sequences.

• In an arithmetic sequence like 3, 6, 9, the common difference is 3 because that’s what you add to each term to get the next.

• In a geometric sequence like 2, 4, 8, the common ratio is 2 because that’s what you multiply each term by to get the next one.

Practice Questions

Question 1:

Write down the first four terms of the arithmetic sequence where the first term is 3 and the common difference is 2.

Solution

The first four terms would be 3, 5, 7, 9

You start with 3 and add the common difference of 2 each time.

---

Question 2:

If the first term in an arithmetic sequence is 5 and the common difference is -3, what would be the fourth term?

Solution

The fourth term would be

---

Question 3:

Write down the first three terms of the geometric sequence where the first term is 2 and the common ratio is 3.

Solution

The first three terms would be 2, 6, 18

You start with 2 and multiply by 3 each time.

---

Question 4:

What would be the 5th term in the geometric sequence that starts with a first term of 1 and has a common ratio of 2?

Solution

The 5th term would be