Geometric Sequences
💡If the gaps keep growing, check for a constant ratio instead of a constant difference. Each term is the previous one multiplied by the same number.
Try it — find the next term
liveA constant ratio means each term multiplies by the same number.
- 1
Divide each term by the one before
common ratio = 2
- 2
Multiply the last term by the ratio
24 × 2 = 48
⏱️ Practice against the clock
A worked example
3, 6, 12, 24, __: each term doubles, so the next is 48.
Look for a steady multiplier
When the gaps keep getting wider, stop hunting for a common difference and check for a common ratio. Divide each term by the one before it. In 3, 6, 12, 24 you get 2 every time, so it doubles and 48 is next.
The tell is acceleration. Arithmetic sequences grow by adding the same step; geometric ones grow by multiplying, so they pull away faster and faster.
No need to rebuild it
Once two or three terms agree on the ratio, you are done poking at it. Multiply the last term you can see by that ratio and you have the next one. No reason to reconstruct the whole run from the start.
That matters once the numbers balloon. Nailing the ratio early and applying it once beats tracking every single term.
Questions people ask
How do I tell a geometric sequence from an arithmetic one? ▾
Divide each term by the one before it. If you keep getting the same ratio, it is geometric; if the plain differences match instead, it is arithmetic.
The numbers grow really fast — any shortcut? ▾
Confirm the ratio from the first two or three terms, then just multiply the last visible term by it. You never need to rebuild the whole sequence.