The big idea: Arithmetic = add (or subtract) the same number every step. Always + or −, never × or ÷.
That fixed step is the common difference, d.
Example: 5, 8, 11, 14, … adds 3 each time, so d = 3.
- Sequence
- A list of numbers in a set order. Each number is a term.
- Term (uₙ)
- One number in the sequence. uₙ is the term in position n.
- First term (u₁)
- The term you start from, in position 1.
- Common difference (d)
- The constant gap between one term and the next.
A decreasing sequence: In 20, 17, 14, 11, … each term is 3 less than the one before, so d = −3.
A negative common difference means the sequence decreases.
How to tell it is arithmetic
- Look at the gaps between consecutive terms.
- If every gap is the same → arithmetic.
- 2, 6, 10, 14 has gaps 4, 4, 4 → arithmetic (d = 4).
- 2, 6, 18, 54 has gaps 4, 12, 36 → not arithmetic (that one multiplies).
Find d from the gap: The step d is the same every time — so subtract the two values, then divide by the number of steps between them.
See it: 5, ?, 11
An arithmetic sequence is 5, ?, 11 — the 1st and 3rd terms.
Find the common difference d.
Step by step
- Same d is added each step: 5 → ? → 11.
- Subtract the two values.
- Count the steps between them (3rd − 1st).
- Share that gap across the steps.
Final answer
d = 3, so the sequence is 5, 8, 11.
[Diagram: math-arithmetic-steps] - Available in full study mode
Step it up — terms further apart
The 3rd term is 17 and the 7th term is 41.
Find the common difference.
Step by step
- Subtract the two values.
- Count the steps between them (7th − 3rd).
- Share that gap across the steps.
Final answer
d = 6.
Count steps, not terms: The 3rd and 7th terms are 4 steps apart (7 − 3), not 7. Always divide by the number of steps, not the term numbers.
Memorize terms 3x faster
Smart flashcards show you cards right before you forget them. Perfect for definitions and key concepts.
The nth-term formula — your main tool: One formula jumps straight to any term — without writing them all out.
- the first term
- the common difference
- the position (which term you want)
IB-style question — find any term
An arithmetic sequence has u₁ = 5 and d = 3. Find the 10th term.
Step by step
- Write the formula.
- Substitute u₁ = 5, d = 3, n = 10. You add d only (10 − 1) = 9 times.
- Work it out.
Final answer
u₁₀ = 32.
IB-style question — build the general term from two terms
The 2nd term is 6 and the 5th term is 18.
Find an expression for the nth term uₙ.
Step by step
- Find d first — gap ÷ steps (from the last section).
- Step back to the first term: from u₂, subtract one d.
- Put u₁ = 2 and d = 4 into the formula and tidy up.
Final answer
uₙ = 4n − 2.
Why (n − 1)? Count the +d jumps from the start: The number on d is just how many +d jumps you've made from the start. You begin at u₁ (0 jumps), so the 10th term is only 9 jumps along — not 10. That is the whole reason for the (n − 1).
So always spot u₁ first: whatever you're given first is term 1, with 0 jumps. Count from there.
Counting TERMS → use n − 1
- Term 1 is the start — 0 jumps.
- The nth term is n − 1 jumps along.
- uₙ = u₁ + (n − 1)d
Counting YEARS / STEPS → use n
- "Now" (year 0) is the start — 0 jumps.
- Each year adds d once → n jumps.
- after n years = u₁ + nd
Read it straight off: Sometimes the exam hands you the rule itself (like uₙ = 20 − 4n) instead of u₁ and d — so you read them straight off it, no working.
IB-style question — read u₁ and d from an explicit rule
The nth term of an arithmetic sequence is uₙ = 20 − 4n.
Write down the first term and the common difference.
Step by step
- Common difference: just read off the number in front of n. No working needed.
- First term: put n = 1 into the rule.
- Not convinced d is just the coefficient? Find a second term and subtract — same answer.
Final answer
u₁ = 16 and d = −4.
Why d is the coefficient of n: Expand the formula: uₙ = u₁ + (n − 1)d = dn + (u₁ − d).
The number multiplying n is always d — that is why you can read it straight off.
Know your predicted grade
Take timed mock exams and get detailed feedback on every answer. See exactly where you're losing marks.
Set the term equal and solve: A common exam question gives you a term's value and asks which term it is — set uₙ equal to that value and solve for the position n.
IB-style question — which term equals a value
An arithmetic sequence has u₁ = 4 and d = 5. Which term is equal to 99?
Step by step
- Write down the formula.
- Put the values in (u₁ = 4, d = 5) and set it equal to 99.
- Expand and simplify.
- Solve for n.
Final answer
The 20th term equals 99.
n must be a whole number: A value is only a term if solving gives a positive whole number.
If n comes out as a decimal (like 7.5), the value is not in the sequence.
Equal gaps: A classic exam twist: three terms in a row contain an unknown. Their gaps must be equal, so set u₂ − u₁ = u₃ − u₂ and solve.
IB-style question — find the unknown that makes it arithmetic
The first three terms of an arithmetic sequence are:
u₁ = k + 2
u₂ = 2k + 3
u₃ = 5k − 2
Find the value of k.
Step by step
- Equal differences: u₂ − u₁ = u₃ − u₂.
- Simplify each side.
- Solve for k.
Final answer
k = 3 (the sequence is 5, 9, 13, with d = 4).
Common mistakes
- Setting the terms equal instead of the differences.
- Adding d n times instead of (n − 1) times.
- Taking d as the constant in uₙ = a + bn instead of the coefficient b.
Do this instead
- Set u₂ − u₁ = u₃ − u₂ (the gaps are equal).
- Use uₙ = u₁ + (n − 1)d — the first term is already u₁.
- The number multiplying n is always the common difference.