IB Maths AA Sequences & Series Explained

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Confusing uₙ with Sₙ is a frequent source of lost marks. If a question asks for the value in year 12, it normally requires a term. If it asks for the total accumulated over 12 years, it normally requires a sum.

Arithmetic sequences and series

An arithmetic sequence changes by the same amount between consecutive terms. Its common difference is:

d = uₙ₊₁ − uₙ

The essential formulas are:

Arithmetic exam example

Suppose u₄ = 14 and u₁₀ = 38. Since six equal steps separate the fourth and tenth terms:

6d = 38 − 14, so d = 4.

Using u₄ = u₁ + 3d gives u₁ = 2. Therefore:

S₁₀ = 10/2(2 + 38) = 200.

Notice the mark-producing structure: establish an equation for the difference, find the first term, and then select a sum formula. Writing only calculator outputs may conceal the method that examiners need to see.

Geometric sequences and series

A geometric sequence has a constant multiplier between consecutive non-zero terms. Its common ratio is:

r = uₙ₊₁/uₙ

Geometric exam example

Suppose u₂ = 12 and u₅ = 324. Dividing the term equations gives:

u₅/u₂ = r³ = 27, so r = 3.

Since u₂ = u₁r, the first term is u₁ = 4. The sum of the first five terms is therefore:

S₅ = 4(1 − 3⁵)/(1 − 3) = 484.

A common error is to write r² rather than r³. Always count the number of transitions between the two term positions.

Infinite geometric series

An infinite geometric series has a finite sum only when |r| < 1. This condition includes negative ratios, which produce alternating positive and negative terms.

For example, consider:

18 − 6 + 2 − 2/3 + ...

Here, u₁ = 18 and r = −1/3. Since |−1/3| < 1, the series converges and:

S∞ = 18/[1 − (−1/3)] = 13.5.

Do not use the infinite-sum formula merely because a series contains an ellipsis. You must first state or verify that |r| < 1. The RevisionDojo infinite geometric sequence resources provide focused practice on this decision.

Sigma notation

Sigma notation represents a sum compactly. In

Σ from k = 1 to 5 of (3k + 2),

k is the index, 1 is the lower limit, 5 is the upper limit and 3k + 2 generates each term. Expanding gives:

5 + 8 + 11 + 14 + 17.

For an arithmetic series, identify the first generated term and the difference. For a geometric series, identify the first generated term and the ratio. Be careful when the lower limit is not 1, because the first term must be found by substituting the actual lower limit.

IB questions may ask you to write a sum using sigma notation, expand a sigma expression, or evaluate the sum. These are different tasks. If asked to write sigma notation, include the expression, index and both limits.

Financial applications and exponential change

Compound interest and annual depreciation are geometric models because the quantity is repeatedly multiplied by the same factor.

A balance of $500 growing by 6% annually is modelled by 500(1.06)ⁿ after n years. To find when it first exceeds $800, solve:

500(1.06)ⁿ > 800.

This gives n > ln(1.6)/ln(1.06) ≈ 8.06, so the first whole number of years is 9. The final rounding is contextual: eight years is insufficient, so ordinary rounding to eight would be wrong.

Depreciation works similarly. A 12% annual decrease uses the multiplier 0.88, not 0.12. Students can review the model in RevisionDojo’s financial applications materials and then watch the corresponding worked financial applications videos.

How examiners phrase sequences and series questions

Questions often hide the sequence inside a context. A theatre adding the same number of seats per row is arithmetic. A population increasing by the same percentage is geometric. Translate the words into u₁, d or r before substituting into a formula.

Under the assessment structure shown in the official IB subject brief, Paper 1 does not allow technology, while Paper 2 allows technology. HL students also sit a technology-allowed Paper 3. Sequences may appear in short-response, extended-response or problem-solving questions, so students need both exact algebraic methods and efficient calculator use.

The HL connection: proof by induction

At HL, proof by mathematical induction can be applied to statements about sequences and sums. A complete induction proof normally has three stages:

1. Verify the statement for the initial value.
2. Assume it is true for n = k.
3. Use that assumption to prove it for n = k + 1, then state the conclusion.

The induction hypothesis must actually be used in the third stage. Merely substituting k + 1 into the proposed formula does not prove the result. This is additional HL proof content, not a different set of arithmetic and geometric formulas.

Common mistakes that cost marks

• Using n instead of n − 1 in an nth-term formula
• Treating a percentage increase as an arithmetic addition
• Finding a term when the question asks for a total
• Using S∞ without checking |r| < 1
• Rounding a threshold answer in the wrong direction
• Losing the negative sign when r is negative
• Entering a formula into a calculator without identifying u₁, d or r
• Giving a numerical answer without units or contextual interpretation

Keep exact values through the working and round only the final answer unless the question instructs otherwise. If a question says “show that,” write enough algebra to establish the stated result rather than citing a calculator display.

An efficient revision method

First, learn to classify a model as arithmetic, geometric, finite or infinite. Next, practise identifying the parameters before touching the formula booklet. Finally, complete mixed questions so that the required method is not announced in advance.

The RevisionDojo arithmetic sequences Questionbank and geometric sequences Questionbank support targeted practice. After attempting each problem independently, use the arithmetic worked video solutions and geometric worked video solutions to see how the method is organized line by line.

Watching a solution is most useful after making a genuine attempt. Compare the setup, formula choice and communication rather than only the final value. Jojo AI can then help identify whether an error came from the underlying concept, algebra or interpretation.

Conclusion

IB Maths AA sequences and series become manageable when you can identify the pattern, distinguish a term from a sum, and interpret the result. Arithmetic models use a constant difference, geometric models use a constant ratio, and infinite geometric sums require |r| < 1.

Revision should combine formula fluency with exam-style application. RevisionDojo’s Questionbank and per-question worked video solutions are particularly useful for seeing how these ideas translate into complete, mark-earning methods.

Sources and referenced URLs

• Official IB Mathematics: analysis and approaches guide
• Official IB Mathematics AA subject brief
• IB Mathematics AA formula booklet
• RevisionDojo arithmetic sequences and series Questionbank
• RevisionDojo arithmetic sequences and series videos
• RevisionDojo geometric sequences and series Questionbank
• RevisionDojo geometric sequences and series videos
• RevisionDojo infinite geometric sequence resources
• RevisionDojo financial applications notes
• RevisionDojo financial applications videos

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