Arithmetic vs Geometric Sequences: How to Tell Them Apart

Quick answer: An arithmetic sequence adds the same number every step (the common difference, d). A geometric sequence multiplies by the same number every step (the common ratio, r). To tell them apart, subtract consecutive terms first — if every difference is identical, it is arithmetic. If not, divide consecutive terms; identical ratios mean geometric.

I got this wrong in a mock last year and it still annoys me. The sequence was 4, 12, 36, 108, and I saw "it goes up by 8, then 24" and panicked. I never divided. Once I learned the two-step check, sequences became the fastest marks on the paper.

Subtract first, then divide

The test takes about ten seconds and never fails. Do it in this exact order.

1. Subtract each term from the one after it. Write the differences underneath.
2. If all the differences match, it is arithmetic and that number is d.
3. If not, divide each term by the one before it. If those ratios all match, it is geometric and that number is r.

Try it on 2, 6, 18, 54. Differences: 4, 12, 36 — not constant, so not arithmetic. Ratios: 6/2 = 3, 18/6 = 3, 54/18 = 3 — constant, so geometric with r = 3.

Now try 20, 17, 14, 11. Differences: -3, -3, -3. Arithmetic with d = -3. A shrinking sequence is not automatically geometric, and never judge by how fast the numbers move: 2, 4, 6, 8 and 2, 4, 8, 16 start identically and are completely different.

The formulas you actually need

Five formulas cover almost every sequences question. a is the first term, n is the term number.

• Arithmetic nth term: a + (n - 1)d
• Arithmetic sum of n terms: n/2 x (2a + (n - 1)d), or n/2 x (a + L) when you already know the last term L
• Geometric nth term: a x r^(n-1)
• Geometric sum of n terms: a(r^n - 1)/(r - 1), for r not equal to 1
• Sum to infinity, only when -1 < r < 1: a/(1 - r)

What you need depends on your system. GCSE with AQA, Edexcel or OCR and Cambridge IGCSE stay on nth term rules and quadratic sequences. CBSE class 10 goes deep on arithmetic progressions including the sum. Cambridge International A-Level and AP push into geometric series and sums to infinity. WASSCE uses both progressions heavily, usually with the terms described in words.

The (n - 1) is the part people get wrong. It is there because the first term has had zero steps applied, not one.

Worked example: an arithmetic sequence, start to finish

Sequence: 7, 11, 15, 19, ...

Step 1 — identify. Differences are 4, 4, 4. Arithmetic, a = 7, d = 4.

Step 2 — nth term. a + (n - 1)d = 7 + 4(n - 1) = 7 + 4n - 4 = 4n + 3. Always simplify. The unsimplified version earns a method mark, but the answer mark wants 4n + 3.

Step 3 — find the 20th term. 4(20) + 3 = 83.

Step 4 — is 143 in this sequence? Set 4n + 3 = 143, so n = 35. Because n is a whole number, yes, 143 is the 35th term. Had n come out as 35.5, the answer is no, and "n is not an integer, so 143 is not a term" is the sentence that earns the mark.

Step 5 — sum of the first 20 terms. Using n/2 x (2a + (n - 1)d): 20/2 x (14 + 76) = 10 x 90 = 900. Check with the other version: 20/2 x (7 + 83) = 900. A free way to verify under pressure.

Worked example: a geometric sequence and a sum to infinity

Sequence: 5, 15, 45, 135, ...

Ratios are 3, 3, 3. Geometric, a = 5, r = 3.

8th term: 5 x 3^(8-1) = 5 x 3^7 = 5 x 2187 = 10935. The exponent is 7, not 8, and writing 3^8 is the most common slip in the whole topic.

Sum of the first 8 terms: a(r^n - 1)/(r - 1) = 5(3^8 - 1)/(3 - 1) = 5(6561 - 1)/2 = 5 x 3280 = 16400.

Now a shrinking one: 12, 6, 3, 1.5, ... Ratios are 0.5 each time, so r = 1/2. Sum to infinity = a/(1 - r) = 12/0.5 = 24. You could add terms forever and never pass 24. That only works because r sits between -1 and 1; with r = 3 the terms keep growing and the sum has no limit. Stating that condition is often worth a mark on its own.

When the exam gives you terms instead of the sequence

Papers rarely hand you a neat list. A typical phrasing is: the 3rd term of an arithmetic sequence is 11 and the 7th term is 27. Find a and d.

Write both as equations: a + 2d = 11 and a + 6d = 27. Subtract the first from the second: 4d = 16, so d = 4. Substitute back: a + 8 = 11, so a = 3. The sequence is 3, 7, 11, 15, 19, 23, 27, which checks out.

The same trick works for geometric sequences, except you divide the equations instead of subtracting. If the 2nd term is 6 and the 4th is 54, then ar = 6 and ar^3 = 54. Dividing gives r^2 = 9, so r = 3 or r = -3, and both count unless the question says the terms are positive. To check working like this line by line, the math solver shows each step.

What if it is neither?

If neither the differences nor the ratios are constant, check the second differences. Take 3, 8, 15, 24, 35. First differences: 5, 7, 9, 11. Second differences: 2, 2, 2. A constant second difference means a quadratic sequence.

Halve the second difference to get the coefficient of n^2. Here 2/2 = 1, so the rule starts with n^2. Subtract n^2 from each term: 3 - 1 = 2, 8 - 4 = 4, 15 - 9 = 6, 24 - 16 = 8. That leftover is 2n, so the nth term is n^2 + 2n. Test with n = 4: 16 + 8 = 24. Correct.

Mistakes that cost marks

• Using n instead of n - 1 in the exponent or the difference.
• Calling a fast-growing sequence geometric without dividing to check.
• Using the sum to infinity formula when r is outside -1 to 1.
• Rounding r early, then feeding that value into a sum. Keep fractions.
• Writing only the final number. Method marks live in the substitution line, so show a + (n - 1)d with your numbers in it first.

Test yourself

1. Is 81, 27, 9, 3 arithmetic or geometric, and what is d or r?
2. Find the nth term and the sum of the first 15 terms of 6, 10, 14, 18, ...
3. A geometric sequence has first term 8 and r = 1/4. What is its sum to infinity?

Answers: geometric with r = 1/3; nth term 4n + 2, sum = 15/2 x (12 + 14 x 4) = 7.5 x 68 = 510; sum to infinity = 8/(1 - 0.25) = 32/3, about 10.67.

FAQ

How do I tell arithmetic from geometric quickly in an exam?

Subtract consecutive terms first: constant differences mean arithmetic. If the differences vary, divide consecutive terms and look for a constant ratio. Subtraction is easier in your head, which is why it goes first.

What is the difference between a sequence and a series?

A sequence is an ordered list, like 2, 5, 8, 11. A series is the sum of those terms, 2 + 5 + 8 + 11. Questions asking for "the nth term" want the sequence rule; "the sum of the first n terms" wants the series formula.

Can a sequence be both arithmetic and geometric?

Yes, but only a constant non-zero sequence like 5, 5, 5, 5, which has d = 0 and r = 1 at once. Exams rarely ask, but it is a nice detail to know.

Do I need the sum to infinity formula for GCSE?

No. GCSE with AQA, Edexcel or OCR sticks to nth terms and quadratic sequences. Sum to infinity turns up in Cambridge International A-Level and AP work. Check your specification first, then use a quick quiz to see which types you can already handle.

In short

Subtract, then divide. Constant difference means arithmetic, so use a + (n - 1)d. Constant ratio means geometric, so use a x r^(n-1). Constant second difference means quadratic, so halve it and subtract n^2. Write the substitution line before the answer, and check whether the question wants a term or a sum. If a step still feels shaky, get it broken down on the explain page or practise more on the maths hub.