Sequences & Series (AI HL)
A sequence is an ordered list of numbers following a rule; a series is what you get when you add the terms of a sequence together. This topic covers the two structures IB tests most - arithmetic sequences, where you add a fixed amount each step, and geometric sequences, where you multiply by a fixed ratio each step - along with the formulas for finding a given term, summing a finite number of terms, and (for geometric series only) summing infinitely many terms.
What the syllabus says
This topic maps onto three points in the official IB Applications & Interpretation syllabus.
| Code | Syllabus content |
|---|---|
| SL1.2 | Arithmetic sequences and series. Use of the formulae for the \(n\)th term and the sum of the first \(n\) terms. Use of sigma notation for sums of arithmetic sequences. Applications, including simple interest over a number of years, and analysis/interpretation where a real-life model is not perfectly arithmetic. |
| SL1.3 | Geometric sequences and series. Use of the formulae for the \(n\)th term and the sum of the first \(n\) terms. Use of sigma notation for sums of geometric sequences. Applications such as the spread of disease, salary increase/decrease, and population growth. |
| AHL1.11 | The sum of infinite geometric sequences. Links to the concept of a limit. |
SL1.2 and SL1.3 are core AI content examined at both SL and HL; AHL1.11 (sum to infinity) is HL-only.
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 an arithmetic sequence?
An arithmetic sequence is a list of numbers where you get from one term to the next by adding a fixed amount, the common difference \(d\). The \(n\)th term is \(u_n = u_1 + (n-1)d\), where \(u_1\) is the first term.
e.g. \(u_1=8,\ d=5\): the 25th term is \(u_{25}=8+24(5)=128\).
What is a geometric sequence?
A geometric sequence is a list of numbers where you get from one term to the next by multiplying by a fixed ratio, the common ratio \(r\). The \(n\)th term is \(u_n = u_1 r^{n-1}\).
e.g. \(u_1=5,\ r=2\): the 11th term is \(u_{11}=5(2)^{10}=5120\).
What is sigma notation?
Sigma notation, \(\sum\), is shorthand for adding up a sequence of terms without writing every one out. The number below \(\Sigma\) is where the count starts, the number above is where it stops, and the expression to the right is the rule for each term.
e.g. \(\displaystyle\sum_{n=1}^{5} 3n = 3(1+2+3+4+5) = 45\).
What is the sum of a geometric series?
The sum of the first \(n\) terms of a geometric sequence is \(S_n = \dfrac{u_1(r^n-1)}{r-1}\) for \(r\neq1\). It works for any finite number of terms, however large.
e.g. \(u_1=2,\ r=3,\ n=4\): \(S_4 = \dfrac{2(3^4-1)}{3-1} = \dfrac{2(80)}{2} = 80\).
What is the sum to infinity?
The sum to infinity, \(S_\infty = \dfrac{u_1}{1-r}\), is the limiting value that the partial sums of a geometric series approach as you add more and more terms - but only when \(|r|<1\), so each term shrinks towards zero.
e.g. \(u_1=6,\ r=0.5\): \(S_\infty = \dfrac{6}{1-0.5} = 12\).
Key formulas
Five formulas cover almost every question on this topic. The two tables below summarise all of them at a glance - the explanations underneath go into more depth on each one.
Formula reference
All five formulas below are on the official formula booklet, so you don't need to memorise them - you need to recognise which one applies and substitute correctly.
| Formula | Used for | Booklet? |
|---|---|---|
| \(u_n = u_1 + (n-1)d\) | Arithmetic \(n\)th term | ✓ Yes |
| \(S_n = \dfrac{n}{2}\big(2u_1+(n-1)d\big)\) | Arithmetic sum of \(n\) terms | ✓ Yes |
| \(u_n = u_1 r^{n-1}\) | Geometric \(n\)th term | ✓ Yes |
| \(S_n = \dfrac{u_1(r^n-1)}{r-1},\ r\neq1\) | Geometric sum of \(n\) terms | ✓ Yes |
| \(S_\infty = \dfrac{u_1}{1-r},\ |r|<1\) | Sum to infinity | ✓ Yes |
Arithmetic vs geometric
Both sequence types share the same underlying idea - a first term plus a rule for stepping to the next one - but the rule and the resulting formulas differ completely.
| Feature | Arithmetic | Geometric |
|---|---|---|
| Step rule | Add \(d\) each time | Multiply by \(r\) each time |
| \(n\)th term | \(u_n=u_1+(n-1)d\) | \(u_n=u_1r^{n-1}\) |
| Sum of \(n\) terms | \(S_n=\tfrac{n}{2}(2u_1+(n-1)d)\) | \(S_n=\tfrac{u_1(r^n-1)}{r-1}\) |
| Sum to infinity? | Never converges (unless \(d=0\)) | Converges when \(|r|<1\) |
| Typical example | Salary rising by a fixed amount each year | Population growing by a fixed percentage each year |
Arithmetic sequences & series
Every arithmetic problem comes down to identifying \(u_1\) and \(d\), then substituting into one of the two formulas above.
\(n\)th term
\[u_n = u_1+(n-1)d\]
Gives any individual term directly, without listing the ones before it.
✓ In the formula bookletSum of \(n\) terms
\[S_n = \tfrac{n}{2}\big(2u_1+(n-1)d\big)\]
Adds the first \(n\) terms without having to sum them one by one.
✓ In the formula bookletSigma notation
\[\sum_{k=1}^{n} u_k = S_n\]
Sums an arithmetic sequence using \(\Sigma\) - identify the first term and common difference from the general term.
Not in the formula booklet - notation onlyGeometric sequences & series
The same pattern applies with the common ratio \(r\) in place of \(d\) - but the sum formula only works when \(r\neq1\), and the sum-to-infinity formula only exists when the series actually converges.
\(n\)th term
\[u_n = u_1 r^{n-1}\]
Gives any individual term directly from the first term and the ratio.
✓ In the formula bookletSum of \(n\) terms
\[S_n = \dfrac{u_1(r^n-1)}{r-1}\]
Adds the first \(n\) terms of a geometric sequence, for any \(r\neq1\).
✓ In the formula bookletSum to infinity
\[S_\infty = \dfrac{u_1}{1-r},\quad |r|<1\]
The limit the partial sums approach as \(n\to\infty\) - only defined when the terms shrink towards zero.
✓ In the formula bookletWorked 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 \(8\) and common difference \(5\).
(a) Find the 25th term.
(b) Find the 50th term.
Worked solution
(a) \(u_{25}=u_1+24d=8+24(5).\) M1
\(u_{25}=8+120=128.\) A1
(b) \(u_{50}=u_1+49d=8+49(5).\) M1
\(u_{50}=8+245=253.\) A1
A geometric sequence has first term \(5\) and common ratio \(2\).
(a) Find the sum of the first 10 terms.
(b) Find the 11th term.
Worked solution
(a) \(S_{10}=5\dfrac{2^{10}-1}{2-1}.\) M1
\(S_{10}=5(1023)=5115.\) A1
(b) \(u_{11}=5(2)^{10}.\) M1
\(u_{11}=5(1024)=5120.\) A1
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 \(d\) and \(r\). Using the arithmetic formulas on a geometric sequence (or vice versa) is the single most common error - check whether consecutive terms have a constant difference or a constant ratio before picking a formula.
- Applying the sum-to-infinity formula when \(|r|\geq1\). \(S_\infty\) only exists when the terms are shrinking towards zero. If \(r=1.2\), for example, the series diverges and there is no sum to infinity, however tempting the formula looks.
- Off-by-one errors in \(n\). \(u_1\) is the first term, not the "zeroth" term, so the 25th term uses \((n-1)=24\), not \(25\). The same trap catches out sigma-notation bounds.
- Losing track of which quantity a recursive step produces. In recurrence relations like \(u_{n+1}=0.6u_n+8\), it's easy to report \(u_n\) when the question asked for \(u_{n+1}\), or to stop iterating one step early.
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.
Geometric sums grow fast - \(2^{50}\) has 16 digits. Scientific notation avoids typing long strings of zeros and prevents rounding errors when a sum or term gets large.
- Scientific notation means \(a\times10^n\), e.g. \(3.2\times10^8\) or \(4.5\times10^{-3}\).
- Use 2nd → , (EE) to enter the ×10 part: type 3.2 2nd , 8 to enter 3.2×10⁸. Do NOT type ×10^ separately.TI-84
- Use the EE key (or type ×10^ from the keyboard template) to enter scientific notation. Or just type 3.2×10^8 using the ^ key.Nspire
- Use the ×10ˣ key (EXP key) - type 3.2 then EXP then 8. Do NOT type ×10^ manually.Casio
- To display answers in scientific notation: on TI-84 press MODE and choose SCI; on Casio set the display mode in SET UP.
Tip: A common mistake is typing ×10^ instead of using the EE/EXP key - this gives ×10×... (multiplication, then a power) rather than proper scientific notation.
Useful for questions like "find the least \(n\) for which \(S_n>1000\)" - faster and safer than trial substitution, and it finds every solution, not just one.
- Graph \(f(x)\) first so you can see how many solutions exist and roughly where they are.
- Rearrange so everything is on one side: \(f(x) = 0\) - or graph both sides as separate functions and find intersections.
- MATH → Solver: enter the expression, type a starting guess close to one root, press ALPHA + ENTER. Move the guess to near a different root and repeat for each solution.TI-84
- Type nSolve(f(x)=0, x, guess) - include a guess or interval e.g. nSolve(f(x)=0, x, 2) or nSolve(f(x)=0, x, {1,5}) to target a specific root.Nspire
- Run-Matrix → SolveN(f(x), x) returns all real roots at once; or use the Equation app for a visual approach.Casio
- For transcendental equations (e.g. \(e^x=3x\)), graph both sides, count crossings, then use the intersection tool for each one.
- Always verify each solution by substituting back into the original equation.
Tip: The solver finds ONE root near your starting guess - change the guess to find others. The graph shows you how many to expect.
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 common difference between consecutive terms (you add the same number each time). A geometric sequence has a constant common ratio (you multiply by the same number each time). Different structure, different nth-term and sum formulas.
When does a geometric series have a sum to infinity?
Only when the common ratio \(r\) satisfies \(|r|<1\). If \(|r|\geq1\) the terms don't shrink towards zero, so the partial sums grow without bound and there is no finite sum to infinity.
Are sequences and series examined with a calculator?
Yes - this is entirely calculator-permitted content in AI. You're expected to use your GDC to evaluate large sums, iterate recursive sequences in a table, and solve for an unknown term or number of terms numerically.
How do I know whether to use the sum formula or just list terms?
List terms for small, simple cases where a pattern is obvious or you only need a couple of terms. Use the sum formula (or your GDC's sequence tools) whenever \(n\) is large, or the question explicitly asks for a sum, since listing dozens of terms by hand wastes time and invites arithmetic slips.
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 AI HL syllabus unit, in case you want to keep going.