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Geometric 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.
📈 Multiply by the same number every step. Formula book: tₙ = t₁ × r(n − 1), where r is the common ratio. r > 1 grows, 0 < r < 1 shrinks (decay).

What it is

A geometric sequence multiplies by the same number each step, instead of adding. That multiplier is the common ratio, r. Doubling bacteria, halving medication, investments growing, radioactive decay, all geometric.

The formula (formula book)

tₙ = t₁ × r(n − 1)
tₙ
the nth term
t₁
the first term
r
common ratio

To find r, divide any term by the one before: r = t₂ ÷ t₁.

📈 r > 1 → growth
gets bigger each step (e.g. r = 2 doubles)
📉 0 < r < 1 → decay
gets smaller each step (e.g. r = 0.5 halves)

Worked example

A post is shared 3 times on day 1, and the shares triple each day. So t₁ = 3, r = 3. How many shares on day 5?

  1. tₙ = t₁ × rn−1 → t₅ = 3 × 35−1 = 3 × 34.
  2. 34 = 81, so t₅ = 3 × 81.
  3. = 243 shares.

Which day first hits 729? Set tₙ = 729: 3 × 3n−1 = 729 → 3n−1 = 243 = 35 → n = 6.

Watch out

• It's rn−1, not rn. At n = 1 the power is 0, and r0 = 1, so t₁ = t₁ × 1. ✓
• r < 1 means decay (shrinking), not a mistake.
• To find r, divide consecutive terms (geometric), don't subtract (that's arithmetic).
• The formula book writes the first term as t₁, not a.

Each bar triples (r = 3)

2 6 18 54 162 ×3 ×3 ×3 ×3

2, 6, 18, 54, 162 … each term is 3 times the last. To find r: 6 ÷ 2 = 3, 18 ÷ 6 = 3 … always the same.

Growth vs decay

📈 r = 2 (growth)
5, 10, 20, 40 … doubling
📉 r = 0.5 (decay)
800, 400, 200, 100 … halving

Bigger than 1 climbs, between 0 and 1 falls. Same formula either way.

Warm up first

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

Sequence 2, 6, 18, 54 … what is r?
r = 6 ÷ 2 = 3. Divide consecutive terms.
Is r = 0.5 growth or decay?
Decay, it's between 0 and 1, so the terms shrink (halving).
t₁ = 5, r = 4. What is t₃?
5 × 43−1 = 5 × 4² = 5 × 16 = 80.

Faded example: shares tripling (t₁ = 3, r = 3)

Rung 1 · watch one done fully

Day 5: t₅ = 3 × 35−1 = 3 × 34 = 3 × 81 = 243.

Rung 2 · you fill the gaps

Day 4: t₄ = 3 × 34−1 = 3 × 33 = 3 × ? = ?

Check my gaps
3 × 27 = 81.
Rung 3 · all you

A medication starts at 800 mg and halves every hour (r = 0.5). How much is left after 3 hours (the 4th reading, t₄)? Check below.

Check my answer
t₄ = 800 × 0.54−1 = 800 × 0.53 = 800 × 0.125 = 100 mg.

Exam-style stretch: find r from two terms

A geometric sequence has t₁ = 7 and t₄ = 189. Find the common ratio r.

Show the working
t₄ = t₁ × r³ → 189 = 7 × r³ → r³ = 189 ÷ 7 = 27 → r = ∛27 = 3. (Check: 7, 21, 63, 189 ✓.)

Say it back

In one sentence, out loud: how do you tell a geometric sequence from an arithmetic one?

⚡ Geometric sequences, one look

Patternmultiply by the same number each step
Formulatₙ = t₁ × rn−1
Common ratio rr = t₂ ÷ t₁ (divide consecutive terms)
r > 1growth · 0 < r < 1 decay
Powerrn−1 not rn (n = 1 gives r⁰ = 1)
Examplet₁=3, r=3 → t₅ = 3 × 3⁴ = 243
Trapdivide (not subtract) to get r · it's rn−1