Sequences (Edexcel GCSE Maths): Revision Notes

Sequences

What is the nth term?

When working with sequences, you'll often need to find a formula called the "nth term" or "general term". This is essentially an expression that allows you to calculate any term in the sequence by substituting different values of n (the position number). Think of it as a mathematical recipe that works for every term in the sequence.

The nth term is particularly useful because it gives you a direct way to find any term without having to list out all the previous terms.

Finding the nth term of an arithmetic sequence

Arithmetic sequences are sequences where you add (or subtract) the same number each time to get from one term to the next. This constant value is called the common difference. There are two main approaches you can use to find the nth term formula for these sequences.

Method 1: Work it out step by step

This method involves systematically working through the pattern to build up your formula. Here's how it works:

The process involves three key steps:

1. Find the common difference - Look at how much the sequence increases (or decreases) each time. This tells you what to multiply n by in your formula.

2. Work out the adjustment needed - Once you know what to multiply n by, you need to figure out what to add or subtract to get the correct terms.

3. Combine everything together - Put both parts together to create your complete nth term formula.

Method 2: Learn the standard formula

For arithmetic sequences, there's a standard formula you can memorise and apply directly:

$\text{nth term} = dn + (a - d)$

Where:

• $d$ is the common difference

• $a$ is the first term

Working with quadratic sequences

Not all sequences are arithmetic. Some follow quadratic patterns, where the nth term involves $n^2$. These require a different approach but the same checking principle applies.

With quadratic sequences, you're given the formula and asked to find specific terms or determine whether certain values appear in the sequence.

Deciding if a term is in a sequence

Sometimes you'll be given a formula and asked whether a specific number appears in the sequence. Here's the strategy:

1. Set up an equation - Make the nth term formula equal to the value you're testing

2. Solve for n - Work out what value of n would give you that term

3. Check if n is a whole number - If n is a positive whole number, the value is in the sequence. If not, it isn't.

Checking your work

Always verify your answers by substituting values back into your formula. This is one of the most important habits you can develop when working with sequences. If your formula gives you the correct terms when you substitute $n = 1$, $n = 2$, etc., you can be confident it's right.

For checking whether a term is in a sequence, make sure your arithmetic is correct when solving the equation. Double-check your calculations, especially when dealing with square roots or fractions.