# Generalise Number Patterns

## Generalising Linear Number Patterns

We can use algebra to create a general rule for a number sequence. For linear sequences, this is very easy. A linear sequence is a number sequence that goes up by the same amount every time. So for example 1, 4, 7, 10 (+3). The advantage of finding a rule for a number sequence is that we can determine any number in that sequence. This is how we do it. Look at the first few numbers of the sequence and see by how much it goes up or down each time. Then use this value and put it in front of n. In the example, this is 3. “n” stands for the term in the sequence (eg: 2 is the first number in the sequence and 5 the second in the example). Now calculate any number in the sequence using xn (3n in this example). There might be a difference between the actual number and the number that you get using your formula. Modify the formula accordingly (in the example, we had to take away 1). Now you can find any number in the sequence. The 100th number in this sequence would be 299.
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Example 1 In this number sequence, the numbers increase by two each time. Consequently, we must write down 2n. However, we need to add 2 to make the formula work.
Example 2 In this number sequence, the numbers increase by four each time. Consequently, we must write down 4n. However, we need to subtract 3 to make the formula work.

## Generalising Quadratic Number Patterns

Generalising quadratic number patterns can be a bit more challenging as generalising linear ones. Quadratic number patterns do not go up or down by the same amount every time. An example of a quadratic number sequence is: 3, 6, 12 & 18. However, it is still possible to generalise these types of sequences by finding the second difference as shown in the example. First find the first difference as we did with the linear sequences and then solve the second difference (the difference of the differences). Here is a rule for the second differences: If the 2nd difference is 2, your formula starts with n² If the 2nd difference is 4, your formula starts with 2n² If the 2nd difference is 6, your formula starts with 3n² In our example, the second difference is two, so our formula starts with n². Then we need to modify the formula to make it work. In this case, we had to add 3. These formulas can be more complicated to modify (eg: n² + 3n -1 or n² +2n). You need to use your maths skills and common sense to change the formula so that works. These number patterns can get very complicated. You can even have n³ depending on the sequence. However, by knowing how to generalise linear and quadratic number sequences you are at a pretty decent level.

## Presentation

Example 1 In this number sequence, the second difference is +4. Consequently, the first part of the formula is 2n². We had to add 2n in this example in order for the formula to work.
Example 2 In this number sequence, the second difference is +6. Consequently, the first part of the formula is 3n². We had to subtract 1 in this example in order for the formula to work.

## More Algebra

You should try to do some linear equations now as this is another basic algebra skill that you should be able to master. If you want to have a look at all our topics, make sure to visit our library. Please share this page if you like it or found it helpful!
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Chapter 11.0:  Learning Outcomes Students will be introduced to number patterns! Students will learn how to generalise linear and quadratic number patterns!

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