A linear sequence goes up or down by the same amount each time, and its nth term has the form dn + c. You meet this skill whenever a question asks for “the nth term” or for a term far along the list, such as the 40th.
It opens the module on sequences and pattern rules and is used again in every other lesson here.
How does a constant difference become a rule?
If each term is d more than the one before, then moving from term n to term n + 1 adds d. So the nth term must contain d × n. That part of the rule says how fast the sequence grows.
The constant c is the correction. It is whatever you must add or subtract so that dn lands on the real terms.
How to find the rule, step by step
- Subtract each term from the next to find the common difference d. Check that it is the same every time.
- Write dn: the multiples of d starting from n = 1.
- Compare dn with the real terms. The gap, with its sign, is the constant c.
- Write the rule as dn + c.
- Check with n = 1 and one later term.
Worked example
Find the nth term of 9, 14, 19, 24, … and then the 10th term.
Step 1, difference: 14 − 9 = 5, 19 − 14 = 5, 24 − 19 = 5. So d = 5.
Step 2, multiples: 5n gives 5, 10, 15, 20.
Step 3, compare: 9 − 5 = 4, 14 − 10 = 4, 19 − 15 = 4. Every term is 4 more than 5n, so c = 4.
Step 4, rule: the nth term is 5n + 4.
Step 5, check: n = 1 gives 9 and n = 4 gives 24. Both match.
10th term: 5 × 10 + 4 = 54. Counting on, 9 + 9 × 5 = 54 as well.
The constant 4 is the “zeroth term”, the value the sequence would have one step before it starts: 9 − 5 = 4.
The mistake to watch for
A frequent slip is to use the first term as the constant.
Mistaken answer: nth term = 5n + 9
The student saw that the sequence starts at 9 and wrote 9 at the end.
Test it: n = 1 gives 5 × 1 + 9 = 14, but the first term is 9. The rule is one whole step too high. The first term already includes the first step, so the constant must be the first term minus d: 9 − 5 = 4.
Check yourself
Try these without a calculator, then open each answer.
1. Find the nth term of 5, 8, 11, 14, …
Show answer
d = 3, and 3n gives 3, 6, 9, 12. Each real term is 2 more, so the rule is 3n + 2.
Check: n = 1 gives 5 and n = 4 gives 14. Both match.
2. Find the nth term of 20, 17, 14, 11, …
Show answer
d = −3, and −3n gives −3, −6, −9, −12. Each real term is 23 more, so the rule is −3n + 23.
Check: n = 1 gives 20 and n = 4 gives 11. Both match.
3. The sequence 6, 10, 14, 18, … continues. Find the 100th term.
Show answer
d = 4, and 4n gives 4, 8, 12, 16, so the constant is 2 and the rule is 4n + 2. The 100th term is 4 × 100 + 2 = 402.
Check by counting on: 6 + 99 × 4 = 6 + 396 = 402.
Where this leads next
Once you can find a rule reliably, learn to test a term rule against distant terms, then try the sequences practice set. The sequence and series laboratory lets you generate terms from a rule so you can check your own.
Some students get the difference right and still lose the constant under time pressure. That is the sort of step a teacher can catch in online one-to-one Mathematics tuition.