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Arithmetic sequences
Same idea, four ways to study it. Tap a style and find the one that clicks for you. 📺 Prefer to watch? Videos are on the lesson page.
📐 Add the same amount every step. Formula book: tₙ = t₁ + (n − 1)d, where t₁ is the first term, d is the common difference, n is the term number.

What it is

An arithmetic sequence adds (or subtracts) the same amount to get from one term to the next. That fixed step is the common difference, d. Like a training plan adding 3 km a week, or cinema rows with 3 more seats each.

The formula (formula book)

tₙ = t₁ + (n − 1)d
tₙ
the nth term
t₁
the first term
d
common difference

To find d, subtract any term from the next: d = t₂ − t₁. It can be negative (a decreasing sequence).

Worked example

Cinema: row 1 has 12 seats, each row has 3 more. So t₁ = 12, d = 3. How many seats in row 10?

  1. Write the formula: tₙ = t₁ + (n − 1)d.
  2. Substitute n = 10: t₁₀ = 12 + (10 − 1) × 3 = 12 + 27.
  3. = 39 seats.

To find which row has 42 seats, set tₙ = 42 and solve: 12 + (n−1)×3 = 42 → n = 11.

Watch out

• It's (n − 1) lots of d, not n. The first term already exists, you add d zero times to reach t₁.
• d can be negative, watch the sign.
• "Find the term" → substitute n. "Which term equals X" → set tₙ = X and solve for n.
• The formula book writes the first term as t₁, not a.

A staircase with equal steps

3 7 11 15 19 +4 +4 +4 +4

3, 7, 11, 15, 19 … every step up is exactly +4. That constant step is the common difference d = 4.

What the formula is really saying

Start at t₁, then add d, but only (n − 1) times, because the first term is already there.

To reach the 10th term: start, then add d nine times.

Warm up first

Don't read yet, just have a go in your head:

Sequence 5, 9, 13, 17 … what is d?
d = 9 − 5 = 4. The constant step.
To reach the 6th term, how many times do you add d?
5 times (n − 1 = 6 − 1).
t₁ = 10, d = −2. What is t₃?
10 + 2×(−2) = 10 − 4 = 6. (d is negative, so it decreases.)

Faded example: cinema seating (t₁ = 12, d = 3)

Rung 1 · watch one done fully

Row 10: t₁₀ = 12 + (10 − 1) × 3 = 12 + 27 = 39 seats.

Rung 2 · you fill the gaps

Row 7: t₇ = 12 + (7 − 1) × 3 = 12 + ? = ?

Check my gaps
12 + 18 = 30 seats.
Rung 3 · all you

The sequence 5, 9, 13, 17 … Write the general term tₙ, then find t₂₀. Check below.

Check my answer
t₁ = 5, d = 4 → tₙ = 5 + (n − 1)×4 = 4n + 1. So t₂₀ = 4(20) + 1 = 81.

Exam-style stretch: two terms

The 4th term of an arithmetic sequence is 22 and the 9th term is 47. Find d and t₁.

Show the working
From t₄ to t₉ is 5 steps of d: 47 − 22 = 25, so d = 25 ÷ 5 = 5. Then t₄ = t₁ + 3d = 22 → t₁ + 15 = 22 → t₁ = 7.

Say it back

In one sentence, out loud: why is it (n − 1) lots of d and not n?

⚡ Arithmetic sequences, one look

Patternadd the same amount each step
Formulatₙ = t₁ + (n − 1)d
t₁first term
dcommon difference = t₂ − t₁ (can be negative)
Find a termsubstitute n
Which term = Xset tₙ = X, solve for n
Examplet₁=12, d=3 → t₁₀ = 12 + 9×3 = 39
Trap(n − 1) lots of d, not n · watch the sign of d