Sequences & the nth Term — Year 9 Revision Notes
These notes cover term-to-term rules, finding and using the nth term of a linear sequence, and checking whether a number is a term — all at Year 9 (Stage 9) level.
Term-to-term rules
A linear sequence goes up or down by the same amount each time, called the common difference. The term-to-term rule describes this step: for 5, 8, 11, 14 the rule is 'add 3'. This is useful for continuing a sequence but not for jumping to a distant term.
Key Facts & Formulas
- Common difference = term − previous term
- 5, 8, 11, 14 → add 3
Tips
- Find the common difference first.
- A decreasing sequence has a negative common difference.
Finding the nth term
The nth term of a linear sequence is (common difference) × n + a constant. Find the common difference for the n term, then adjust with a constant to match the first term. For 4, 7, 10, 13: difference 3 gives 3n, and 3n at n = 1 is 3, so add 1 → nth term = 3n + 1.
Key Facts & Formulas
- nth term = dn + c
- 4, 7, 10, 13 → 3n + 1
Tips
- The common difference is the coefficient of n.
- Check your rule by substituting n = 1, 2, 3.
Using the nth term
Once you have the nth term, substitute a position number to find that term — the 10th term of 3n + 1 is 3×10 + 1 = 31. To check if a number is in the sequence, set the nth term equal to it and see if n is a whole number.
Key Facts & Formulas
- 10th term of 3n + 1 = 31
- Solve 3n + 1 = k for whole n
Tips
- Substitute the position number, not the term value.
- If n is not a whole number, the value is not in the sequence.
Revision Checklist
- I can continue a linear sequence and state its term-to-term rule
- I can find the nth term of a linear sequence
- I can use the nth term to find any term
- I can check whether a number is a term of a sequence
Frequently Asked Questions
Is 100 a term of the sequence 3n + 1?
Set 3n + 1 = 100, so 3n = 99 and n = 33. Because n is a whole number, 100 is the 33rd term of the sequence. If n had not been a whole number, 100 would not be in the sequence.
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