Arithmetic Sequences (AI SL)
Each term in an arithmetic sequence differs from the last by the same fixed amount - the common difference - which is why the sequence's graph of value against position is a straight line. Here you'll find the nth-term and sum formulas laid out side by side, worked examples, and the mistakes that cost the most marks. It's part of the broader Sequences & Series topic.
49 questions on this sub-topic.
The two formulas
Covered under IB syllabus reference SL1.2. Both formulas come from the formula booklet, so you don't need to memorise them - but you do need to recognise which one a question wants.
nth term
\(u_n = u_1 + (n-1)d\)
Use when you need one specific term - "find the 10th term", "which term first exceeds 100?".
Sum of n terms
\(S_n = \dfrac{n}{2}(2u_1 + (n-1)d) = \dfrac{n}{2}(u_1 + u_n)\)
Use when you need a total - "find the sum of the first 40 terms". The second form is faster once you already know \(u_n\).
Need the full syllabus wording and formula-booklet reference table? See Sequences & Series.
Worked examples
In an arithmetic sequence the 3rd term is 11 and the 7th term is 27.
(a) Find the common difference.
(b) Find the first term.
Worked solution
(a) Each step adds \(d\), and \(u_7\) is \(4\) steps past \(u_3\). M1
So \(u_7 - u_3 = 4d.\) A1
\(4d = 27 - 11 = 16 \Rightarrow d = 4.\) A1
(b) Back-substitute into \(u_3 = u_1 + 2d\): \(11 = u_1 + 2(4).\) M1
\(\Rightarrow u_1 = 11 - 8 = 3.\) (Check: \(u_7 = 3 + 6(4) = 27.\)) A1
Find the sum of the first 40 terms of the arithmetic sequence \(3, 8, 13, \dots\)
Worked solution
Arithmetic with \(u_1 = 3\) and common difference \(d = 8-3 = 5.\) A1
\(S_n = \tfrac{n}{2}(2u_1 + (n-1)d)\) (best when \(u_n\) is not yet known): \(S_{40} = \tfrac{40}{2}(2(3) + 39(5)) = 20(6 + 195) = 20(201)\) M1 \(= 4020.\) A1
Common mistakes
- Mixing up \(d\) and \(r\). A constant difference means arithmetic (\(d\)); a constant ratio means geometric (\(r\)). Confusing the two applies the wrong pair of formulas entirely.
- Using \(n\) instead of \((n-1)\). The nth-term formula multiplies \(d\) by \((n-1)\), not \(n\) - an easy off-by-one error under time pressure.
- Solving inequalities without rounding sensibly. "Which term first exceeds 100" gives a decimal value of \(n\) - the answer must be rounded up to the next whole term, then checked against the sequence itself, not just rounded to the nearest integer.
Ready to practise properly?
49 arithmetic-sequence questions, marked instantly like the real exam.
Quick answers
What is the formula for the nth term of an arithmetic sequence?
\(u_n = u_1 + (n-1)d\), where \(u_1\) is the first term and \(d\) is the common difference.
What is the formula for the sum of an arithmetic series?
\(S_n = \tfrac{n}{2}(2u_1 + (n-1)d)\), or equivalently \(S_n = \tfrac{n}{2}(u_1 + u_n)\) once you know the last term.
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.