Number patterns and sequences are core aspects of the GCSE Maths curriculum. Spotting links between numbers is a crucial mathematical skill that can even help us to understand many natural phenomena. Nature “behaves” mathematically and many different patterns can be found. Of these, the pattern described by the Fibonacci sequence is probably the most famous. In this sequence of numbers each term after the second is formed by adding the two previous terms:

The Italian mathematician Leonardo Fibonacci discovered it in 1202 whilst he was investigating the breeding patterns of rabbits. Since then, it has been found that the progression described by this sequence is present in many natural shapes such as in the seeds heads of sunflower plants.

## Linear sequences

There are different types of sequences you’ll encounter at a GCSE Maths level. A linear or arithmetic sequence has the same difference between a term and the next. For instance, in the sequence 2, 5, 8, 11, 14, …, the difference is 3.

Broadly speaking, we would like to find a rule that can summarise all the numbers of a sequence without having to write them down. Such rules are known as the ** nth term**, and can be expressed using the formula

**. We know that n =**

*An ± b**1, 2 3, 4,…, thus we only need to find*

*A*and

*b*.

*A*is the difference between numbers, so in the above sequence,

**.**

*A*= 3To find *b,* we simply need to calculate the difference between the first term and *A*. In this case, *b* = 2 – 3, hence ** b = -1.** Consequently,

**the**.

*n*th term is equal to 3*n*– 1You can always check your work by substituting *n = *1, 2 3, 4,… into the *n*th term.

Let’s now have a look at a more complicated example of a GCSE sequences question.

**Given the sequence 5, 12, 19, 26, 33, …., find the ***n*th term, the 25th term and the first term that is greater than 500.

*n*th term, the 25th term and the first term that is greater than 500.

The difference between consecutive numbers is 7, thus *A = 7 *and *b = 5 – 7* which is equal to -2 . Consequently, **the nth term is equal to 7n – 2.** Then, the 25th term can be found by substituting

*n*= 25 into the rule 7

*n*– 2. Hence, the 25th term = 7 x 25 which is equal to 173. To find the first term greater than 500, we need to solve the following inequality:

7*n *– 2 > 500

7*n > *500 + 2

*n* > 71.7

Thus, the first integer greater than 500 is the 72nd. This term can be found by substituting into the *n*th rule:

** 7 x 72 – 2 = 502**.

## Geometric sequence

A sequence in which you can find each term by multiplying the previous term by a fixed value is called a **geometric sequence**. The *n*th term of a geometric sequence is given by ** a x rⁿ⁻¹,** where

*a*is the first term and

*r*is the multiplier.

### For instance, take the sequence 2, 6. 18. 54. 162, …., *a *= 2 and *r = *3. This means that the *n*th term equals 2 x 3*ⁿ*⁻¹.

At this point you might want to try to find the nth term of the sequence formed by prime numbers. Well, the global scientific community is still actively looking for a solution to this problem!

## TuitionWorks is here to help

If you or your child is still feeling less than confident about sequences, or any other aspects of the GCSE maths syllabus, TuitionWorks can provide an intensive course of personalised, one-to-one lessons with a qualified teacher like me. Just get in touch for a free consultation.

### Federico Antonelli

Maths tutor at TuitionWorks

I am an applied mathematician and qualified secondary teacher. I have done research in the field of nuclear energy and am currently studying toward a PhD with the University of Cranfield in aerospace materials.