Arithmetic Sequences (AI HL)
A sequence is arithmetic when a constant amount - the common difference - is added (or subtracted) to move from one term to the next, giving a linear pattern of growth. Below are the nth-term and sum formulas, real worked examples, and the mistakes most likely to cost marks. It's part of the broader Sequences & Series topic.
15 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 20th term", "find the first term greater than 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 20 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
A sequence is \(5, 9, 13, \dots\)
(a) Find the 20th term.
(b) Find the sum of the first 20 terms.
Worked solution
(a) \(u_1 = 5,\ d = 4\): \(u_{20} = 5 + 19(4)\) M1
\(= 81.\) A1
(b) \(S_{20} = \tfrac{20}{2}(5 + 81)\) M1
\(= 860.\) A1
Evaluate \(\displaystyle\sum_{k=1}^{15}(4k - 1)\).
Worked solution
This is arithmetic: \(u_1 = 3,\ u_{15} = 4(15)-1 = 59,\ n\) M1
\(n = 15.\) A1
\(S = \tfrac{15}{2}(3 + 59)\) M1
\(S = 465.\) A1
A person saves $50 in month 1 and increases the monthly amount by $10 each month. After how many months do total savings first exceed $5000?
Worked solution
Arithmetic: \(S_n = \tfrac{n}{2}(100 + 10(n-1))\) M1
\(= 5n^2 + 45n.\) A1
Solve \(5n^2 + 45n > 5000 \Rightarrow n^2 + 9n - 1000 > 0.\) M1 A1
Root \(n \approx 27.0\), so \(n\) A1
\(= 28\) months. A1
Common mistakes
- Mixing up \(d\) and \(r\). Using the geometric formulas on an arithmetic 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.
- 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.
- Forgetting sigma notation is still an AP. \(\sum_{k=1}^{15}(4k-1)\) is a sum of an arithmetic sequence in disguise - identify \(u_1\), \(d\), and \(n\) first, then use the normal sum formula rather than trying to expand term by term.
Ready to practise properly?
18 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.