Sequences in Context (AI SL)
Exam questions rarely say the word "arithmetic" or "geometric" outright - they describe a salary, a theatre, a bouncing ball, or a fundraiser, and expect you to spot the pattern yourself. This page is about that first step: reading a real-world scenario, deciding which sequence model fits, and then applying the right formula. It's part of the broader Sequences & Series topic.
12 questions on this sub-topic.
Spotting the model
Covered under IB syllabus references SL1.2 (arithmetic) and SL1.3 (geometric). Both sets of formulas are in the formula booklet - the exam skill being tested here is choosing between them from a written description.
Constant amount → arithmetic
\(u_n = u_1 + (n-1)d,\quad S_n = \dfrac{n}{2}(2u_1+(n-1)d)\)
A flat rise or fall each step: "$1500 more each year", "4 extra seats each row".
Constant factor → geometric
\(u_n = u_1 r^{n-1},\quad S_n = \dfrac{u_1(r^n-1)}{r-1}\)
A percentage change or repeated fraction each step: "10% more", "reaches 60% of the previous height".
Need the full syllabus wording and formula-booklet reference table? See Sequences & Series.
Worked examples
A charity runs a series of 6 fundraising events. The first event raises $500 in donations. Due to growing publicity, each subsequent event raises 1.2 times as much as the event before it.
Find the total amount raised over all 6 events, correct to the nearest dollar.
Worked solution
Donations form a geometric sequence with \(u_1=500,\ r=1.2,\ n=6\): \(S_6=\dfrac{u_1(r^n-1)}{r-1}=\dfrac{500(1.2^6-1)}{1.2-1}\) M1
\(=\dfrac{500(1.985984)}{0.2}=4964.96\) A1
\(\approx$4965.\) A1
A theatre has 24 seats in row 1 and 4 more in each successive row, for 18 rows.
(a) Seats in the last row.
(b) Total number of seats.
Worked solution
(a) Seats in the last row. Arithmetic with \(u_1 = 24,\ d = 4\): \(u_{18} = 24 + 17(4)\) M1
\(= 92.\) A1
(b) Total seats. \(S_{18} = \dfrac{18}{2}(24+92) = 9(116)\) M1
\(= 1044.\) A1
Common mistakes
- Guessing the model instead of testing it. "Grows by 10%" is a ratio, not a difference - check whether consecutive values share a constant difference or a constant ratio before picking a formula, rather than assuming from the wording alone.
- Miscounting how many terms the context covers. "Over the first 6 events" or "18 rows" fixes \(n\) directly from the story - it's easy to use one too many or too few terms, especially when the first term is described separately from "each additional" one.
- Not rounding to match the context. Money answers usually need rounding to the nearest cent or dollar, and seat/people counts must be whole numbers - carry full precision through the calculation and round only at the final answer.
Ready to practise properly?
11 context-style sequence questions, marked instantly like the real exam.
Quick answers
How do I know if a word problem is arithmetic or geometric?
Look at how consecutive terms are related. A fixed amount added or subtracted each time (a flat pay rise, seats increasing by 4 per row) is arithmetic. A fixed percentage or multiplying factor (10% growth, a ball bouncing to 60% of its height) is geometric.
Which formula do I use for a running total in a context question?
Once you have identified the model, use the matching sum formula: \(S_n = \tfrac{n}{2}(2u_1+(n-1)d)\) for arithmetic or \(S_n = \dfrac{u_1(r^n-1)}{r-1}\) for geometric, with \(n\) set to however many terms the context describes.
Where can I check whether my GDC is set up correctly for this topic?
See the GDC guidance on the full Sequences & Series page for how to build and sum a sequence list on the TI-84, Casio, and TI-Nspire.