Finding the nth term of a sequence in KS3 involves identifying the pattern and expressing it as a formula. Here’s a breakdown of how to do it:
Understanding the Basics
Before diving in, let’s clarify some key concepts:
- Sequence: A sequence is a list of numbers that follow a specific pattern.
- Term: Each number in the sequence is called a term.
- nth Term: The nth term is a formula that allows you to find any term in the sequence, given its position (n). For example, if n = 1, it gives you the first term; if n = 2, the second term, and so on.
Steps to Finding the nth Term
-
Identify the Pattern: Look at the differences between consecutive terms in the sequence.
- Constant Difference: If the difference is the same, the sequence is linear. This is the most common type at KS3 level. For example, 2, 4, 6, 8... each term has a difference of +2
- Non-constant Difference: If the difference isn't the same, look for other patterns such as squaring, cubing or differences between the differences.
-
Find the 'Multiplier'
- If the difference is constant, this becomes the number multiplied by n in our formula. In our example (2, 4, 6, 8, ...) the difference is 2, so this value is 2.
-
Adjust for the Starting Point: Compare the sequence to the times table associated with the multiplier you found in step 2.
- In our example, we're using the 2 times table because the difference between terms is 2 (2, 4, 6, 8, ...) We would compare to the sequence 2, 4, 6, 8, 10,... because the 2 times table starts at 2.
- If it matches, then you have your formula. However, most sequences at KS3 require an adjustment.
- For example, if our sequence was actually 5, 7, 9, 11,... our multiplier would still be 2, because the difference is constant at 2, however, we must adjust it as it doesn't match the 2 times table.
- The two times table is 2, 4, 6, 8, 10,.... To get from 2 to 5, 4 to 7, 6 to 9, we must add 3. Therefore the nth term is 2n + 3.
-
Write the formula
- Combine the multiplier (from step 2) and any adjustment (step 3) to make your nth term formula.
- Use the n for the term number to make the formula applicable to all terms. So a formula will typically look like an + b where a is the constant difference and b is the adjustment
Examples with solutions
| Sequence | Differences | Times Table | Adjustment | nth Term Formula |
|---|---|---|---|---|
| 3, 5, 7, 9, ... | +2 | 2n | +1 | 2n + 1 |
| 6, 12, 18, 24, ... | +6 | 6n | +0 | 6n |
| 1, 4, 9, 16, ... | +3, +5, +7 | n² | +0 | n² |
| 10, 7, 4, 1, ... | -3 | -3n | +13 | -3n + 13 |
Example from the Reference
The reference in the YouTube video states, "it’s formula. It’s emptor miss 6n because compared to the numbers 1 2 3 4 5 what we do to those numbers as we times by 6 to get 6 times table". This indicates that if a sequence was 6, 12, 18, 24,... the nth term would be 6n. This is because the difference between the terms is constant (6), and therefore this becomes the multiplier for n.
Using the Formula
- Once you have the nth term formula, you can use it to find any term.
- For example if our sequence was 2n + 3 and we wanted to find the 10th term, we simply replace n with 10.
- 2 * 10 + 3 = 23
Tips for Success
- Always double-check your formula by plugging in the first few term numbers.
- Practice with different sequences to become more confident.
- Be patient and try to look for a pattern.
- Write down your working steps to help spot mistakes.
By following these steps, you can successfully find the nth term of a sequence at KS3 level and use it to predict any term in the sequence!
