Sequences & Series (AA HL)
Sequences are lists of numbers that follow a pattern; series are the totals you get from adding them up. This topic covers the two types tested at IB - arithmetic (where each term increases by a fixed amount) and geometric (where each term is multiplied by a fixed amount) - the formulas for finding any term or summing them, sigma notation as shorthand for writing a sum, and what happens when a geometric series is added up forever.
What the syllabus says
This topic maps onto four points in the official IB Analysis & Approaches syllabus, quoted here word for word so you know exactly what's examinable.
| Code | Syllabus content |
|---|---|
| SL1.2 | Arithmetic sequences and series. Use of the formulae for the nth term and the sum of the first n terms of the sequence. Use of sigma notation for sums of arithmetic sequences. Applications. |
| SL1.3 | Geometric sequences and series. Use of the formulae for the nth term and the sum of the first n terms of the sequence. Use of sigma notation for the sums of geometric sequences. Applications. |
| SL1.4 | Financial applications of geometric sequences and series: compound interest; annual depreciation. Calculate the real value of an investment with an interest rate and an inflation rate. |
| SL1.8 | Sum of infinite convergent geometric sequences. Use of \(|r|<1\) and modulus notation. |
These are core AA SL syllabus points that are also examinable at AA HL.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is a sequence?
A sequence is an ordered list of numbers, called terms, where each term follows a consistent rule linking it to the one before. Sequences can be infinite (continuing forever) or finite (stopping after a fixed number of terms), and the two types tested throughout this topic are arithmetic and geometric sequences.
e.g. \(10, 15, 20, 25, 30, \ldots\) is a sequence (add 5 each time) - but \(10, 8, 5, 9, \ldots\) isn't, since there's no consistent rule connecting the terms.
What is a series?
A series is what you get when you add together the terms of a sequence, rather than listing them separately. Like sequences, a series can be finite (a fixed number of terms) or infinite (continuing forever) - and, as the "convergent" definition below shows, only some infinite series actually add up to a finite total.
e.g. for the sequence \(10, 15, 20, 25, 30\), the series is \(10+15+20+25+30 = 100\).
What is the common difference in an arithmetic sequence?
The common difference, \(d\), is the fixed number added to (or subtracted from) each term of an arithmetic sequence to produce the next one. Because it's constant throughout the sequence, knowing \(d\) and the first term lets you find any term without listing every one in between.
e.g. in \(5, 8, 11, 14, \ldots\), each term increases by 3, so \(d=3\).
What is the common ratio in a geometric sequence?
The common ratio, \(r\), is the fixed number each term of a geometric sequence is multiplied by to reach the next one. Unlike a common difference, \(r\) can make a sequence grow explosively (\(r>1\)), shrink towards zero (\(0<|r|<1\)), or alternate in sign (\(r<0\)).
e.g. in \(3, 6, 12, 24, \ldots\), each term doubles, so \(r=2\).
What does convergent mean for a series?
A geometric series is convergent when its terms shrink towards zero fast enough that adding infinitely many of them still settles on a finite total, rather than growing without bound. This happens precisely when the common ratio satisfies \(|r|<1\).
e.g. \(8, 4, 2, 1, \ldots\) has \(r=\tfrac12\), so it converges, and the sum to infinity is 16.
Key formulas
Six formulas cover every question on this topic. The two tables below summarise all of them at a glance - the explanations and worked derivations underneath go into more depth on each one.
Formula reference
All six formulas, side by side, so you can check which ones you actually need to memorise - four are on the official formula booklet, so you only need to actively recall the other two.
| Formula | Used for | Booklet? |
|---|---|---|
| \(u_n = u_1 + (n-1)d\) | Arithmetic - nth term | ✓ Yes |
| \(S_n = \tfrac{n}{2}\big(2u_1 + (n-1)d\big)\) | Arithmetic - sum of n terms | ✓ Yes |
| \(S_n = \tfrac{n}{2}(u_1 + u_n)\) | Arithmetic - sum via first & last term | Not in booklet |
| \(u_n = u_1 r^{\,n-1}\) | Geometric - nth term | ✓ Yes |
| \(S_n = \dfrac{u_1(r^n-1)}{r-1}\) | Geometric - sum of n terms | ✓ Yes |
| \(S_\infty = \dfrac{u_1}{1-r}\) | Sum to infinity (\(|r|<1\)) | ✓ Yes |
Arithmetic vs geometric
The single question that decides which formulas to use: do consecutive terms share a common difference, or a common ratio? This table lines up every other difference that follows from that one choice.
| Feature | Arithmetic | Geometric |
|---|---|---|
| Rule | Add \(d\) each term | Multiply by \(r\) each term |
| Growth pattern | Linear (straight line) | Exponential (curved) |
| nth term | \(u_n = u_1+(n-1)d\) | \(u_n = u_1 r^{\,n-1}\) |
| Sum of n terms | \(S_n = \tfrac{n}{2}(2u_1+(n-1)d)\) | \(S_n = \dfrac{u_1(r^n-1)}{r-1}\) |
| Sum to infinity? | Never (terms don't shrink to 0) | Yes, if \(|r|<1\) |
| Example | \(5, 8, 11, 14, \ldots\) | \(3, 6, 12, 24, \ldots\) |
Arithmetic sequences
An arithmetic sequence increases (or decreases) by the same fixed amount, \(d\), every term - e.g. \(5, 8, 11, 14, \ldots\) (\(d=3\)). The terms grow in a straight line, which is why arithmetic sums use the "average of first and last term, times how many terms" idea.
Arithmetic - nth term
\[u_n = u_1 + (n-1)d\]
Where \(u_1\) is the first term and \(d\) is the common difference.
✓ In the formula bookletArithmetic - sum of n terms
\[S_n = \tfrac{n}{2}\big(2u_1 + (n-1)d\big)\]
Use this when you know the first term and common difference.
✓ In the formula bookletArithmetic - sum via first & last term
\[S_n = \tfrac{n}{2}(u_1 + u_n)\]
A useful shortcut when you already know the last term \(u_n\).
Not in the formula booklet - derive it if neededGeometric sequences
A geometric sequence multiplies by the same fixed factor, \(r\), every term - e.g. \(3, 6, 12, 24, \ldots\) (\(r=2\)). Because the terms grow (or shrink) exponentially rather than in a straight line, geometric sequences behave very differently: if \(|r|<1\) the terms shrink towards zero and the sum settles on a finite value even when added forever - the sum to infinity.
Geometric - nth term
\[u_n = u_1 r^{\,n-1}\]
Where \(u_1\) is the first term and \(r\) is the common ratio.
✓ In the formula bookletGeometric - sum of n terms
\[S_n = \dfrac{u_1(r^n - 1)}{r - 1}, \quad r \neq 1\]
Works for any \(r \neq 1\), including \(r > 1\) or \(r < -1\).
✓ In the formula bookletSum to infinity
\[S_\infty = \dfrac{u_1}{1 - r}, \quad |r| < 1\]
Only valid when the series converges, i.e. \(-1 < r < 1\).
✓ In the formula bookletSigma notation
\(\Sigma\) (sigma) is shorthand for "add up all these terms" - it saves writing out a long sum by hand. The number below \(\Sigma\) is where the counter starts, the number above is where it stops, and the expression to the right is what gets added each time.
Reading \(\displaystyle\sum_{r=1}^{5}(2r+1)\)
Substitute \(r = 1, 2, 3, 4, 5\) into \(2r+1\), then add the results:
\[3+5+7+9+11 = 35\]
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
An arithmetic sequence has first term \(u_1 = 5\) and common difference \(d = 3\).
(a) Find \(u_{10}\).
(b) Find the sum of the first 10 terms, \(S_{10}\).
Worked solution
(a) \(u_{10} = u_1 + (n-1)d = 5 + (10-1)(3) = 32\) M1A1
(b) \(S_{10} = \tfrac{10}{2}\big(2(5) + 9(3)\big) = 185\) M1A1
A geometric sequence has first term \(u_1 = 12\) and common ratio \(r = \tfrac{1}{3}\).
(a) Find \(u_5\).
(b) Explain why the series converges, and find \(S_\infty\).
Worked solution
(a) \(u_5 = u_1 r^{\,n-1} = 12\left(\tfrac13\right)^4 = \tfrac{4}{27}\) M1A1
(b) \(|r| = \tfrac13 < 1\), so the series converges. R1 \(S_\infty = \dfrac{12}{1 - \tfrac13} = 18\) M1A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Mixing up \(n\) and \(n-1\). The \(n\)th-term formulas use the exponent/multiplier \((n-1)\), not \(n\) - it's easy to be off by one term, especially under exam pressure.
- Using \(S_\infty\) when \(|r| \geq 1\). The sum to infinity formula only applies to a convergent geometric series. If \(|r| \geq 1\) the series has no finite sum - say so instead of forcing the formula.
- Confusing arithmetic and geometric. Check whether consecutive terms share a common difference (arithmetic) or a common ratio (geometric) before picking a formula - mixing them up gives a completely wrong answer.
- Rounding \(r\) too early. When \(r\) is a decimal derived from other information, keep it exact (as a fraction or unrounded decimal) until the final answer, or small errors compound.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
Evaluate sigma notation, list the terms of a sequence, or total a series without adding by hand.
- Write the general term, e.g. \(u_n = 3n - 1\), and the range of \(n\).
- List terms with 2nd → STAT (LIST) → OPS → 5:seq(expr, n, start, end); total with 2nd → STAT → MATH → 5:sum(seq(...)).TI-84
- Type seq(expr, n, start, end) to list and sum(seq(...)) to total, or use the Σ template from the maths templates.Nspire
- OPTN → LIST → Seq( builds the list; OPTN → LIST → Sum totals it.Casio
- For Σ notation the lower and upper limits are the start and end values of \(n\).
Tip: Many calculators have a Σ( template - enter the term, the variable, and the two limits directly.
Advance a recurrence - a loan balance each month, a population each year - without retyping the previous value.
- Write the rule, e.g. next balance \(= \text{balance} \times 1.02 - 200\), and the starting value.
- Type the start value, ENTER; then the rule using 2nd → (−) (ANS), e.g. Ans×1.02−200, and press ENTER repeatedly.TI-84
- Enter the first value, then a formula referring to the previous answer; press ENTER repeatedly.Nspire
- In Run-Matrix enter the start value, then the rule using Ans, and press EXE repeatedly.Casio
- Each ENTER/EXE gives the next term - count the presses to reach the term you need.
Tip: This Ans-recursion trick works for any step-by-step process, not just sequences.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Sequences & series questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What's the difference between an arithmetic and a geometric sequence?
An arithmetic sequence has a constant difference between consecutive terms (you add the same number, \(d\), each time). A geometric sequence has a constant ratio between consecutive terms (you multiply by the same number, \(r\), each time).
When can I use the sum to infinity formula?
Only for a geometric series where the common ratio satisfies \(|r| < 1\). If \(|r| \geq 1\) the series diverges and has no finite sum, so the formula \(S_\infty = \tfrac{u_1}{1-r}\) does not apply.
Is sigma notation examined at HL?
Yes. Sigma notation is used throughout AA HL to write sums of arithmetic and geometric series concisely, and can appear on both Paper 1 and Paper 2.
Can I use my GDC for this topic?
Yes, on Paper 2 - your calculator's sequence/table mode can generate and check terms, and list-sum functions can verify a computed \(S_n\). On Paper 1 you'll need the formulas above without a calculator. See the GDC guide for model-specific instructions.
Sub-topics
Sequences & Series broken down into its individual skills, each with its own focused page.
Related topics
More Number & Algebra topics from the same AA HL syllabus unit, in case you want to keep going.