Arithmetic vs Geometric Sequences: Rules & Practice

Name the position and the term

A sequence is an ordered list. The term at position n is aₙ. In this guide n starts at 1, so the first term is a₁=a. If a source starts at n=0, its formula will look different.

A short list can fit many rules. State the intended pattern or assumptions before extending it. Three attractive numbers alone do not guarantee a unique next term.

Test the differences

In 5,8,11,14, each term increases by 3. This is arithmetic, with a=5 and d=3. A negative difference gives a decreasing arithmetic sequence, such as 12,8,4,0.

Calculate each adjacent difference in order. A changing gap means the sequence is not arithmetic, even if every term is larger than the previous one.

Build the arithmetic nth term

From the first term to the nth term there are n−1 jumps. Therefore aₙ=a+(n−1)d. For the first example, aₙ=5+3(n−1)=3n+2.

Substitute n=1 as a quick check. The incorrect formula a+nd would move one jump too far. If given two non-adjacent terms, divide their difference by the number of position intervals.

Test the ratios

In 3,6,12,24, dividing a term by its predecessor gives 2 each time. This is geometric with a=3 and r=2. The pattern 16,8,4,2 has ratio 1/2.

A negative ratio makes signs alternate. For 2,−6,18,−54, r=−3. Division by zero needs care: do not use a ratio test across a zero denominator or assume a sequence of zeros determines a unique ratio.

Build the geometric nth term

Starting with a, multiply by r once to get the second term and n−1 times to get term n. Thus aₙ=arⁿ⁻¹. For 3,6,12,24, the rule is 3·2ⁿ⁻¹.

A fraction ratio and a negative ratio still fit the same rule. Evaluate the exponent first and keep negative bases in brackets, for example 2(−3)ⁿ⁻¹.

Find a missing term with an independent check

If an arithmetic sequence has term 2 equal to 9 and term 5 equal to 21, there are three position intervals, so d=(21−9)/3=4. The first term is 5.

For a geometric sequence with positive terms 4, __,36, the common positive ratio is 3 and the missing term is 12. If negative ratios are permitted, −12 also works as a different sequence. State the condition rather than silently discarding a branch.

Use pictures and real contexts carefully

An arithmetic tile pattern can add one fixed row each stage. A geometric counter pattern can triple its previous collection. Count the actual objects in the diagram before matching a formula.

“Increasing by 20%” describes a multiplier of 1.2 on the latest value. “Increasing by 20” describes an additive difference. Context should make the unit and repetition interval clear.

Compare patterns without memorising labels

Use a record with positions, terms, adjacent differences and defined adjacent ratios. Then state a rule, calculate a distant term and check that n=1 recovers the first term.

This guide focuses on classifying and generating terms, not series sums. For total sums in arithmetic contexts, use the related arithmetic-progressions guide. Parents can ask why there are n−1 jumps rather than only checking the numerical answer.

See the reasoning

Worked examples and checks

Try, explain, check

12 questions with explained answers

Try each task before opening its explanation. These are original BananaLearn questions, not official exam or past-paper questions. Where several answers can work, the explanation gives the criteria.

1. Classify 9,13,17,21.

   Check answer and reasoning

   Arithmetic, d=4
   All adjacent differences equal 4.

2. Classify 9,18,36,72.

   Check answer and reasoning

   Geometric, r=2
   All adjacent ratios equal 2.

3. Find the nth term of 4,7,10,…

   Check answer and reasoning

   4+3(n−1), or 3n+1
   There are n−1 steps after term 1.

4. Find its tenth term.

   Check answer and reasoning

   31
   4+9×3=31.

5. Find the fifth term of 2,6,18,…

   Check answer and reasoning

   162
   2×3⁴=162.

6. Find the fourth term of 3,−6,12,…

   Check answer and reasoning

   −24
   Multiply by −2 each time.

7. Find the sixth term of 32,16,8,…

   Check answer and reasoning

   1
   32×(1/2)⁵=1.

8. Is 1,4,9,16 arithmetic?

   Check answer and reasoning

   No
   Its differences 3,5,7 are not constant.

9. Is the same sequence geometric?

   Check answer and reasoning

   No
   Its adjacent ratios are not constant.

10. A sequence adds 6 and starts at 10. Is its nth term 10+6n?

    Check answer and reasoning

    No
    That gives 16 when n=1; use 10+6(n−1).

11. Between positive geometric terms 9 and 81, find a positive middle term.

    Check answer and reasoning

    27
    The positive ratio is 3.

12. Can −27 also be a middle term if negative ratios are allowed?

    Check answer and reasoning

    Yes
    9×(−3)=−27, then ×(−3)=81.

A checking record to use in your notebook

Copy these headings and fill them for one worked task, then for a fresh attempt. Use your own evidence rather than copying the model answer.

Question or sentence

What am I being asked to explain?

Model or evidence

Which diagram, rule or passage detail supports my method?

Work

What steps connect the question to the answer?

Independent check

Can I substitute, recount, trace a path or check the text?

Next attempt

What changes in a fresh example?

Choose the next useful step

• Use arithmetic progressions in word problems
• Compare additive and multiplicative models
• Precalculus topic map

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Sources, level and update

Original BananaLearn explanations, examples, illustrations and practice. Source facts checked 7 November 2026. Suggested levels are learning guides, not a claim that every board or textbook uses the same sequence.

• OpenStax free mathematics textbooks: select the relevant Algebra/Precalculus chapter

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