Arithmetic Sequences (AA HL)
An arithmetic sequence moves by the same fixed step, the common difference, from one term to the next, whether that step is positive, negative, or a fraction. This page covers the two formulas behind every arithmetic question - for one term and for a running total - with worked examples and the errors that lose the most marks at HL. It's part of the broader Sequences & Series topic.
14 questions on this sub-topic.
The two formulas
Covered under IB syllabus reference SL1.2. The nth-term formula is in the formula booklet; the sum formula usually appears as \(S_n = \tfrac{n}{2}(2u_1+(n-1)d)\), with the first-and-last-term version below a useful shortcut once \(u_n\) is already known.
nth term
\[u_n = u_1 + (n-1)d\]
Where \(u_1\) is the first term and \(d\) is the common difference.
✓ In the formula bookletSum of n terms
\(S_n = \tfrac{n}{2}\big(2u_1 + (n-1)d\big) = \tfrac{n}{2}(u_1 + u_n)\)
The first form is in the formula booklet; the second (using the last term directly) is quicker once \(u_n\) is known.
Need the full syllabus wording and formula-booklet reference table, or a refresher on your GDC's sequence tools? See Sequences & Series.
Worked examples
An arithmetic sequence has first term \(7\) and common difference \(-3\).
(a) Find the 20th term.
(b) Find the sum of the first 20 terms.
Worked solution
(a)(i) \(S_n = \tfrac{n}{2}(u_1 + u_n) = \tfrac{n}{2}(5 + 95).\) M1
\(50n = 750.\) A1
\(n = 15.\) A1
(a)(ii) \(u_{15} = u_1 + 14d = 95 \Rightarrow 5 + 14d = 95.\) M1
\(14d = 90.\) A1
\(d = \tfrac{45}{7}.\) A1
An arithmetic series has first term \(u_1 = 5\) and last term \(u_n = 95\). The sum of all \(n\) terms is \(750\).
(a)(i) Find \(n\).
(a)(ii) Find the common difference \(d\).
Worked solution
(a)(i) Use the sum formula: \(S_n = \tfrac{n}{2}(u_1 + u_n) = \tfrac{n}{2}(5 + 95).\) M1
Simplify: \(50n = 750.\) A1
Solve: \(n = 15.\) A1
(a)(ii) Use the nth term formula: \(u_{15} = u_1 + 14d = 95 \Rightarrow 5 + 14d = 95.\) M1
Simplify: \(14d = 90.\) A1
Solve: \(d = \tfrac{45}{7}.\) A1
Common mistakes
- Confusing arithmetic and geometric. Check whether consecutive terms share a common difference (arithmetic) or a common ratio (geometric) before picking a formula - mixing them up gives a completely wrong answer.
- 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, especially when \(n\) is large.
- Ignoring a faster route when \(u_n\) is already known. If the last term is given or found in an earlier part, \(S_n=\tfrac{n}{2}(u_1+u_n)\) avoids re-deriving \(d\) unnecessarily and reduces the chance of arithmetic slips.
Ready to practise properly?
16 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.