Sequences & Series (AA SL)
A sequence is an ordered list of numbers following a rule; a series is what you get when you add its terms together. This topic covers arithmetic sequences (constant difference) and geometric sequences (constant ratio), the formulas for finding any term or any partial sum, the special case of a sum to infinity, and applying all of this to real situations like salaries, savings and bouncing balls.
What the syllabus says
This topic maps onto three points in the official IB Analysis & Approaches syllabus.
| 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. Use of sigma notation for sums of arithmetic sequences. Applications, including modelling situations that are not perfectly arithmetic in real life. |
| SL1.3 | Geometric sequences and series. Use of the formulae for the nth term and the sum of the first \(n\) terms. Use of sigma notation for sums of geometric sequences. Applications, including the spread of disease, salary increase and decrease, and population growth. |
| SL1.8 | Sum of infinite convergent geometric sequences, using \(|r|<1\) and modulus notation. |
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 add (or subtract) the same constant amount, the common difference \(d\), to get from one term to the next. Its graph of term value against position is a straight line.
e.g. \(u_1=4,\ d=5\) gives \(4, 9, 14, 19,\dots\)
What is a geometric sequence?
A geometric sequence is a list of numbers where you multiply by the same constant ratio, \(r\), to get from one term to the next. It grows (or shrinks) faster than an arithmetic sequence, since the multiplier compounds.
e.g. \(u_1=5,\ r=2\) gives \(5, 10, 20, 40,\dots\)
What is the nth term of a sequence?
The nth term, written \(u_n\), is a formula that gives the value of any term directly from its position \(n\), without listing every term before it. For arithmetic it's \(u_n=u_1+(n-1)d\); for geometric it's \(u_n=u_1r^{n-1}\).
e.g. \(u_1=4,\ d=5\): \(u_{12}=4+11(5)=59\).
What is a series?
A series is the sum of the terms of a sequence, up to a certain number of terms, written \(S_n\). It's a single number, not a list - the running total of adding \(u_1+u_2+\dots+u_n\).
e.g. For \(u_1=7,\ d=4\): \(S_5=7+11+15+19+23=75\).
What is a sum to infinity?
A sum to infinity is the finite total a convergent geometric series settles towards as you keep adding terms forever. It only exists when \(|r|<1\), so each new term is smaller than the last and the total stops growing meaningfully.
e.g. \(u_1=2,\ r=0.75\): \(S_\infty=\dfrac{2}{1-0.75}=8\).
Key formulas
Five formulas cover every question on this topic - all of them are given in the IB formula booklet, so the skill is choosing the right one and identifying \(u_1\), \(d\) or \(r\) correctly.
Formula reference
Every formula below appears in the formula booklet - you don't need to memorise them, just recognise which situation each one applies to.
| Formula | Used for | Booklet? |
|---|---|---|
| \(u_n=u_1+(n-1)d\) | nth term, arithmetic | ✓ Yes |
| \(S_n=\dfrac{n}{2}(2u_1+(n-1)d) = \dfrac{n}{2}(u_1+u_n)\) | Sum of first \(n\) terms, arithmetic | ✓ Yes |
| \(u_n=u_1r^{n-1}\) | nth term, geometric | ✓ Yes |
| \(S_n=\dfrac{u_1(r^n-1)}{r-1},\ r\neq1\) | Sum of first \(n\) terms, geometric | ✓ Yes |
| \(S_\infty=\dfrac{u_1}{1-r},\ |r|<1\) | Sum to infinity, geometric | ✓ Yes |
Arithmetic vs geometric
The two sequence types share the same shape of formula, but with a very different growth pattern.
| Feature | Arithmetic | Geometric |
|---|---|---|
| Rule between terms | Add \(d\) | Multiply by \(r\) |
| nth term | \(u_n=u_1+(n-1)d\) | \(u_n=u_1r^{n-1}\) |
| Growth shape | Straight line (linear) | Curve (exponential) |
| Sum to infinity? | Never (grows without bound) | Only if \(|r|<1\) |
| Typical application | Salary rising by a fixed amount | Investment growing by a fixed percentage |
Finding terms and sums
Both sequence types need the same two pieces of information: the first term, and either the common difference or the common ratio.
nth term, arithmetic
\[u_n=u_1+(n-1)d\]
Start at \(u_1\) and add \(d\) a total of \(n-1\) times.
✓ In the formula bookletnth term, geometric
\[u_n=u_1r^{n-1}\]
Start at \(u_1\) and multiply by \(r\) a total of \(n-1\) times.
✓ In the formula bookletSum of first n terms
Arithmetic: \(S_n=\tfrac{n}{2}(u_1+u_n)\). Geometric: \(S_n=\tfrac{u_1(r^n-1)}{r-1}\), \(r\neq1\).
✓ In the formula bookletConvergence and sum to infinity
Only geometric series can have a finite sum to infinity, and only under one condition.
Convergence test
A geometric series converges (has a sum to infinity) if and only if \(|r|<1\) - the terms must shrink towards zero.
Sum to infinity
\[S_\infty=\dfrac{u_1}{1-r}\]
Valid only when \(|r|<1\); if \(|r|\ge1\) the series diverges and no sum to infinity exists.
✓ 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 \(u_1=4\) and common difference \(d=5\).
(a)(i) Find \(u_1\).
(a)(ii) Find \(u_2\).
(a)(iii) Find \(u_3\).
(b) Find \(u_{20}\).
(c) Find \(S_{20}\).
Worked solution
(a)(i) \(u_1=4.\)
(a)(ii) \(u_2=9.\)
(a)(iii) \(u_3=14.\) A1
(b) \(u_{20}=4+19(5).\) M1
\(=99.\) A1
(c) \(S_{20}=\dfrac{20}{2}(4+99).\) M1
\(=10(103)=1030.\) A1
A geometric sequence has \(u_1 = 3\) and \(u_4 = 24\).
(a) Show that the common ratio is \(r = 2\).
(b) Find \(u_8\).
(c) Find \(S_8\).
Worked solution
(a) \(u_4=u_1r^3\Rightarrow24=3r^3.\) M1
\(r^3=\dfrac{24}{3}=8.\) A1
Taking the real cube root, \(r=2,\) as required. AG
(b) \(u_8=u_1r^7=3(2)^7.\) M1
\(=3(128)=384.\) A1
(c) \(S_8=\dfrac{u_1(r^8-1)}{r-1}=\dfrac{3(256-1)}{1}.\) M1
\(=3(255)=765.\) 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 arithmetic and geometric formulas. \(u_n=u_1+(n-1)d\) adds; \(u_n=u_1r^{n-1}\) multiplies - using the wrong one produces a completely different sequence.
- Off-by-one errors in the exponent or multiplier. The \(n\)th term uses \((n-1)\) steps from \(u_1\), not \(n\) steps - \(u_1\) itself needs zero additions or multiplications.
- Applying the sum to infinity formula when \(|r|\ge1\). \(S_\infty=\dfrac{u_1}{1-r}\) only makes sense for a convergent series - always check \(|r|<1\) before using it.
- Forgetting real-life sequences aren't always exact. Modelling questions (like a bouncing ball or a growing population) often need the arithmetic or geometric model applied with rounding and reasonableness in mind, not treated as perfectly exact forever.
Using your GDC
The 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.
Faster and safer than algebra for messy equations - useful for finding which term of a sequence first reaches (or exceeds) a target value.
- Graph \(f(n)\) first so you can see how many solutions exist and roughly where they are.
- Rearrange so everything is on one side: \(f(n) = 0\) - 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(n)=0, n, guess) - include a guess or interval e.g. nSolve(f(n)=0, n, 2) or nSolve(f(n)=0, n, {1,5}) to target a specific root.Nspire
- Run-Matrix → SolveN(f(n), n) returns all real roots at once; or use the Equation app for a visual approach.Casio
- 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.
How do I tell if a sequence is arithmetic or geometric?
Arithmetic sequences add (or subtract) a constant amount, d, each term. Geometric sequences multiply by a constant ratio, r, each term. Check consecutive differences for arithmetic, or consecutive ratios for geometric - if either is the same every time, you've found your sequence type.
When does a geometric series have a sum to infinity?
Only when the common ratio satisfies \(|r| < 1\) - the series then converges to a finite total as more and more terms are added. If \(|r| \ge 1\) the terms don't shrink, so the series diverges and has no sum to infinity.
Are the sequence and series formulas in the formula booklet?
Yes. The nth-term and sum formulas for both arithmetic and geometric sequences, plus the sum to infinity formula, are all listed in the IB formula booklet - you don't need to memorise them, just know which one to use.
Can I use my GDC for sequences and series questions?
Yes, on Paper 2 - you can tabulate a sequence to read off terms, or use the equation solver to find which term reaches a target value. On Paper 1 you'll need the nth-term and sum formulas 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 SL syllabus unit, in case you want to keep going.