Sequences and functions
Using the nth term
A term-to-term rule takes you one step at a time. An nth-term rule jumps straight to any term you like.
Position numbers
Write the sequence in a table, with the POSITION NUMBER above each term. position: 1 2 3 4 5 term: 3 6 9 12 15 The position number is simply where the term sits: the first term is 3, the second is 6, the third is 9.
Spotting the rule
In that sequence, the terms are the 3 times table. Multiply the position number by 3 and you get the term: 1 × 3 = 3 2 × 3 = 6 3 × 3 = 9 So the rule is: term = 3 × position number. Use it to jump anywhere: the 8th term = 3 × 8 = 24. You do not have to work through every term in between.
Writing it in algebra
Let n stand for the position number. Then: nth term = 3 × n, or simply nth term = 3n For the sequence 12, 24, 36, 48, 60, … each term is 12 × its position, so the nth term is 12n. The 20th term is 12 × 20 = 240.
Which rule to use when
TERM-TO-TERM tells you how to get the NEXT term. Good for continuing a sequence a step at a time. NTH TERM tells you any term directly from its position. Far better when the position is large — finding the 100th term one step at a time would take all day.
Worked examples
Find the nth term of 3, 6, 9, 12, 15
- Position 1 gives 3, position 2 gives 6, position 3 gives 9.
- Each term is 3 × its position.
Answer: nth term = 3n
The 8th term of that sequence
- Use the rule directly: 3 × 8.
Answer: 24 — no need to work through the others
The nth term of 12, 24, 36, 48, 60
- Each term is 12 × its position.
Answer: 12n
The 20th term of 12, 24, 36, …
- 12 × 20 = 240.
Answer: 240
The nth term of 5, 10, 15, 20
- Every term is 5 times its position number.
Answer: 5n
Which rule for the 100th term?
- Term-to-term would need 99 steps.
- The nth term gets there in one.
Answer: Use the nth term
Where you see this in real life
- A taxi charging £3 a mile has cost = 3n for n miles — you can price a 40-mile trip without adding up 40 separate miles.
- Seat numbering in rows of 12: the last seat in row n is 12n, so row 15 ends at seat 180.
Key words
- Position number
- where a term sits in the sequence
- nth term
- a rule giving any term from its position number
- Sequence
- a list of numbers following a pattern
- Term-to-term rule
- how to get from one term to the next
- Algebra
- using letters to stand for numbers