The same multiplier every time
One person sends a message to 2 people. Each of them sends it on to 2 more. How many people get it in each round?
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- Round 1: 1 person has the message.
- Round 2: 2 people. Double the last round.
- Round 3: 4 people. Double again.
- Round 4: 8 people. Every round: ×2.
The rule
A geometric progression (GP) multiplies by the same number at every step.
That number is the common ratio, r. Find r by dividing any term by the term before it. The first term is a.
Here tₙ is the term at position n. The power n − 1 counts the steps from the first term.
tₙ = a × rⁿ⁻¹
Take 5, 10, 20, 40. Here a = 5 and r = 10 ÷ 5 = 2. Term 6 has 5 steps. So t₆ = 5 × 2⁵ = 160.
Check: 2⁵ = 32, and 5 × 32 = 160.
Which term is 640?
For 5, 10, 20, 40, … set 5 × 2ⁿ⁻¹ = 640. Divide both sides by 5: 2ⁿ⁻¹ = 128, which is 2⁷. So n − 1 = 7 and n = 8.
| Ratio | What happens | Example |
|---|---|---|
| above 1 | the list climbs | 5, 10, 20, 40 |
| exactly 1 | the list holds still | 5, 5, 5, 5 |
| between 0 and 1 | the list shrinks | 32, 16, 8, 4 |
| below 0 | the signs switch each term | 2, −4, 8, −16 |
Mark the points (n, tₙ) on a graph. Here n goes across and tₙ goes up. For an AP, the points lie on a straight line. A GP with r ≠ 1 and a ≠ 0 differs. Any three of its points are not in line.