Sequences

Imagine you're lining up your favourite toys, one after another. If you always put a red car, then a blue car, then a green car, and then repeat, you've made a sequence! In maths, a sequence is simply a list of numbers that follow a specific rule or pattern.

Think of it like a treasure hunt where each clue (number) tells you how to find the next one. Each number in the list is called a term. So, in the sequence 2, 4, 6, 8..., the first term is 2, the second term is 4, and so on. The 'rule' here is to add 2 to get the next number. It's like counting by twos forever!

Let's say you're saving money for a new video game. Your parents agree to give you £5 on the first day, £10 on the second day, £15 on the third day, and so on. This is a sequence!

• Day 1: £5 (This is your first term)
• Day 2: £10 (Your second term)
• Day 3: £15 (Your third term)

Can you guess how much you'd get on Day 4? If you said £20, you got it! The rule (or common difference) here is that you add £5 each day. This type of sequence, where you add or subtract the same amount each time, is called an arithmetic sequence. It's like climbing stairs where each step is the same height.

Let's break down how to understand and find the next numbers in a sequence:

1. Look for the pattern: Start by comparing the first two numbers. What did you do to get from the first to the second? (Did you add, subtract, multiply, or divide?)
2. Test the pattern: Apply that same action to the second and third numbers. Does it work? If not, go back to step 1 and try a different action.
3. Confirm the rule: Once you find an action that works for at least three numbers, you've likely found the rule (also called the common difference for arithmetic sequences or common ratio for geometric sequences).
4. Apply the rule: Use this rule to find the next numbers in the sequence. It's like having a recipe to bake the next cookie!

Just like there are different flavours of ice cream, there are different types of sequences:

• Arithmetic Sequence: This is the one we saw with your savings. You add or subtract the same number every time to get the next term. This 'same number' is called the common difference. Example: 3, 6, 9, 12... (common difference is +3).
• Geometric Sequence: Here, you multiply or divide by the same number every time. This 'same number' is called the common ratio. Example: 2, 4, 8, 16... (common ratio is ×2). Think of it like a snowball rolling down a hill, getting bigger and bigger by multiplying its size.
• Other Sequences: Some sequences have trickier rules! Like the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8...), where you add the two previous numbers to get the next one. But for IELTS, you'll mostly see arithmetic and geometric patterns.

Here are some traps students sometimes fall into:

• ❌ Mistake 1: Only checking the first two terms. You might see 2, 4... and think the rule is 'add 2'. But what if the next number is 8? Then the rule is 'multiply by 2'! ✅ How to avoid: Always check the pattern for at least three terms to be sure. Look at 2, 4, 8 to see it's multiplication.
• ❌ Mistake 2: Confusing arithmetic and geometric. Thinking that 1, 2, 4, 7... is geometric because the numbers are growing. It's not! (1 to 2 is +1, 2 to 4 is +2, 4 to 7 is +3 – no common ratio). ✅ How to avoid: Clearly identify if you're adding/subtracting (arithmetic) or multiplying/dividing (geometric) the same number each time.
• ❌ Mistake 3: Forgetting negative numbers. Sometimes the 'common difference' can be a negative number, meaning the sequence is decreasing (going down). Example: 10, 7, 4, 1... ✅ How to avoid: Remember that 'adding' a negative number is the same as subtracting. The common difference here is -3.