An arithmetic sequence goes up (or down) by the same amount each step, a geometric sequence multiplies by the same ratio . In AI SL you find terms and sums with the booklet formulas and your GDC, and use them in context: savings, populations, seats in rows, a bouncing ball. First decide whether the situation adds or multiplies, then whether the question wants one term or a total.
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, where is the common difference. If the first row of a theatre has 20 seats and each row has 2 more, row 15 has seats.
Arithmetic series
. The 15 rows above hold seats in total.
Geometric sequences
; find by dividing a term by the one before. Growth by % means , decay by % means . A ball dropped from 2 m that rebounds to 70% of its height reaches m on the third bounce.
Geometric series
for . Saving \$100 in the first month and 5% more each month gives after a year.
Finding n with your GDC
When the unknown is , use a table of values or the equation solver, and remember is a whole number. "When does the total first exceed …" means round up to the first that works, and check it in the table.
Key formulas
Arithmetic sequence In the booklet
Arithmetic series In the booklet
Common difference Learn it
Geometric sequence In the booklet
Geometric series, In the booklet
The IB formula booklet is provided in every exam paper. In the booklet = printed there, so know where to find it; Learn it = not in the booklet, so learn it by heart. All AI SL formulas →
Worked example 1: nth term of an arithmetic sequence · easy
An arithmetic sequence has a first term of and a common difference of . Find .
Solution
1. Recall.
2. Substitute, , .
3. Set up: .
4. Simplify the bracket: .
5. Evaluate: .
2. Substitute, , .
5 steps · 1 to fill in
Examiner tip: Watch the : for the 25th term you add exactly times, not times.
Worked example 2: Total distance travelled by a bouncing ball · medium
A ball is dropped from . After each bounce it rebounds to of its previous height. Calculate the total vertical distance travelled from the moment it is dropped until it hits the ground for the 6th time.
Solution
1. Recognise the initial drop is (down only). The 5 subsequent rebounds are up-and-down each.
2. Identify the rebound sequence: , .
3. Use with .
4. Evaluate: .
5. Total: initial drop (sum of rebounds) .
6. State: total distance .
4. Evaluate: m.
6 steps · 1 to fill in
Examiner tip: Between bounces the ball travels the rebound distance TWICE (up and down), so double the rebound sum. The initial drop is only once, so don't double that.
Worked example 3: Solving for n in an arithmetic series · medium
An arithmetic sequence has and . Find the minimum number of terms required for the sum of the series to strictly exceed .
Solution
1. Recall: .
2. Set up: .
3. Simplify: .
4. Expand: .
5. Solve on the GDC: positive root .
6. Conclude: smallest integer .
6 steps
Examiner tip: Instead of manipulating the quadratic algebraically you can graph and on the GDC and use the intersection tool directly.
Sequences & Series exam-style questions
3 questions from our AI SL bank, written in the style of IB papers, easiest first. One has its full worked solution open here. Open the worked answer, or check your answer in the practice area.
Question 1 · easy · 4 marks · Paper 1
An arithmetic sequence has the following first three terms:
Find the common difference, .
Find the value of the 15th term, .
Find the sum of the first 15 terms of the sequence, .
Using the summation tool on your Graphic Display Calculator, evaluate the following arithmetic series:
Show the values you entered into your calculator.
Full worked solution Free, no sign-in
GDC summation entry: expression , variable , lower limit , upper limit . (M1)
(Check: arithmetic series with , , , so .) (A1)
Sum . A1
Every question here has a full worked solution.See them all →
Examiner's noteSequences: uₙ vs Sₙ vs n, and rounding n up
Where marks are lost: Candidates used the arithmetic formula in a geometric situation, found a sum when a term was needed (or the reverse), wrote u₄₂ for "the term equal to 42", found an extra term, found a nearby term instead of the one asked for, did not round n up to a whole number of days, and listed terms instead of using the formula.
Examiners flagged this in 3 exam series (May 2024, November 2024, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →
Sequences & Series on your calculator
A GDC is allowed on every AI paper. Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire, with a checked worked example:
Are the sequence formulas in the AI SL formula booklet?
Yes: the nth term and the sum of n terms for arithmetic and geometric sequences are all in the AI SL booklet. The skill is deciding which one the context needs.
How do I tell arithmetic from geometric in a word problem?
If the same amount is added each time (2 more seats per row), it is arithmetic. If the same percentage or factor is applied (5% more each month, 70% of the height), it is geometric.
How many IB Maths AI SL sequences and series questions are there?
There are 36 exam-style sequences and series questions in the AI SL question bank (Paper 1: 23 · Paper 2: 13), graded 6 easy, 15 medium, 7 hard, 8 very hard. Every question has a full IB-style mark scheme (M, A and R marks).
Is sequences and series on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 23 · Paper 2: 13. Practise with your GDC — AI papers expect calculator methods throughout.