An arithmetic sequence adds the same amount each time. A geometric sequence multiplies by the same amount each time. The skill is deciding which one you have before you write a rule, and it appears in questions about growth, decay and repeated percentage change.
This lesson sits in the module on sequences and pattern rules and builds on finding a linear term rule.
How do you decide between adding and multiplying?
Run two quick tests on consecutive terms. The difference test subtracts each term from the next. The ratio test divides each term by the one before.
If the differences are all equal, the sequence is arithmetic. If the ratios are all equal, it is geometric. If neither, it is something else.
How to work through it, step by step
- Find the differences between consecutive terms.
- Find the ratios of each term to the one before.
- Decide: constant difference means arithmetic, constant ratio means geometric.
- Write the rule: arithmetic is dn + c, geometric is a × r^(n − 1).
- Check by substituting n = 1 and a later term.
Worked example
The sequence 2, 6, 18, 54, … continues. Decide the type, write the nth term and find the 6th term.
Step 1, differences: 6 − 2 = 4, 18 − 6 = 12, 54 − 18 = 36. These are not equal, so it is not arithmetic.
Step 2, ratios: 6 ÷ 2 = 3, 18 ÷ 6 = 3, 54 ÷ 18 = 3. The ratio is constant, so it is geometric with a = 2 and r = 3.
Step 3, rule: nth term = 2 × 3^(n − 1).
Step 4, check: n = 1 gives 2 × 3^0 = 2 and n = 4 gives 2 × 3³ = 2 × 27 = 54. Both match.
6th term: 2 × 3^5 = 2 × 243 = 486. Continuing the pattern, 54 × 3 = 162 and 162 × 3 = 486 as well.
The mistake to watch for
A common slip is to use the wrong power in the geometric rule.
Mistaken answer: nth term = 2 × 3^n, so the 6th term is 2 × 3^6 = 1458
The student wrote the power as n instead of n − 1.
Test it at n = 1: 2 × 3^1 = 6, but the first term is 2. The first term has not been multiplied by the ratio yet, so the power is one less than the position. The correct rule is 2 × 3^(n − 1), giving 486 for the 6th term.
Check yourself
Try these without a calculator, then open each answer.
1. The sequence is 5, 10, 20, 40, … State the type and find the 7th term.
Show answer
Differences: 5, 10, 20, not constant. Ratios: 2, 2, 2, constant. It is geometric with a = 5 and r = 2.
7th term: 5 × 2^6 = 5 × 64 = 320.
2. The sequence is 100, 90, 80, 70, … State the type and find the nth term.
Show answer
Differences: −10, −10, −10, constant, so it is arithmetic with d = −10. Then −10n gives −10, −20, −30, so the constant is 110.
The nth term is 110 − 10n. Check: n = 1 gives 100 and n = 4 gives 70.
3. Is 1, 4, 9, 16, … arithmetic, geometric or neither? Give evidence.
Show answer
Differences: 3, 5, 7, not constant. Ratios: 4, 2.25, about 1.78, not constant. The sequence is neither. These are the square numbers, n².
Where this leads next
Next, learn to use second differences to investigate a quadratic rule, which covers sequences like the squares. Then use the sequences practice set to mix all the types. The sequence and series laboratory shows both kinds of change side by side.
A sequence question can look simple and still hide the wrong type of change. That is something a teacher in online one-to-one Mathematics tuition can help you screen for.