Sequences and variation - Mathematics IGCSE Study Notes

Imagine you're lining up toys in a row. If you put a small car, then a medium car, then a large car, and then repeat that pattern, you've made a sequence! In maths, a sequence is just a list of numbers that follow a specific rule or pattern.

• Arithmetic Sequence: Think of it like climbing stairs, where each step is the same height. You add the same number every time to get to the next number in the list. For example, 2, 4, 6, 8... (you add 2 each time).
• Geometric Sequence: This is like a snowball rolling down a hill, getting bigger and bigger by multiplying. You multiply by the same number every time to get to the next number. For example, 3, 9, 27, 81... (you multiply by 3 each time).

Now, let's talk about variation. This is about how two things are connected and change together. Think of it like a seesaw: if one side goes up, the other side goes down. Or, if one side gets heavier, it goes down.

• Direct Variation: This is like a friendship where if one friend gets happier, the other friend also gets happier. As one quantity (amount) increases, the other quantity also increases at a steady rate. Or, if one decreases, the other decreases. For example, the more hours you work, the more money you earn.
• Inverse Variation: This is like a tug-of-war. As one quantity increases, the other quantity decreases. For example, the more friends you share a pizza with, the smaller each person's slice becomes.

Let's use an example of saving money, which is a great way to see sequences and variation in action!

Sequence Example: Your Savings Account

Imagine you decide to save $5 every week. You start with $10 in your piggy bank.

1. Week 0 (Start): You have $10.
2. Week 1: You add $5, so you have $10 + $5 = $15.
3. Week 2: You add another $5, so you have $15 + $5 = $20.
4. Week 3: You add another $5, so you have $20 + $5 = $25.

See the pattern? The sequence of your savings is 10, 15, 20, 25... This is an arithmetic sequence because you are adding the same amount ($5) each time.

Variation Example: Time to Clean Your Room

Now, let's think about how long it takes to clean your room. This is an example of inverse variation.

1. You clean alone: It takes you 60 minutes.
2. You and one friend clean (2 people): It might take only 30 minutes (60 minutes / 2 people).
3. You and two friends clean (3 people): It might take only 20 minutes (60 minutes / 3 people).

Here, the 'number of people cleaning' and the 'time it takes to clean' are inversely related. As the number of people increases, the time it takes decreases. They vary inversely!