The 5‑D method
One system for every sequence. Linear ones finish early; quadratic ones use all five.
Write the terms under their positions n = 1, 2, 3… and find the gap between neighbours.
1st differences equal → linear. 2nd differences equal → quadratic.
Quadratic: a = 2nd difference ÷ 2. Linear: the difference itself goes in front of n.
Subtract the n‑part from every term. What's left over gives the rest.
Substitute two values of n. One match can be a fluke!
Which path?
Follow the arrows. The differences tell you which kind of sequence you're dealing with.
d = the difference · zero term = one step back from the 1st term
- a = 2nd difference ÷ 2
- Subtract an² from each term
- The leftover is linear → bn + c
Watch it work
Why halve the 2nd difference?
Look at the plain square numbers 1, 4, 9, 16. Each new square grows by an L‑shape of 3, 5, 7… tiles — and each L is 2 bigger than the last.
So n² has a 2nd difference of 2. That means 2n² has 4, 3n² has 6, and in general an² has 2a.
Working backwards: a = 2nd difference ÷ 2.
Spot the mistake
Four students, four classic traps. Work out what went wrong before you reveal it.
Visual Lab
Change the formula. Watch the dots, the differences and the shape respond.
Guided Crack
Do each step yourself — the app checks as you go. Stuck? Show me fills it in. You can also paste a sequence from your homework.
- Differences
- Decide
- Divide
- Deduct
- Double‑check
Levels
Six questions per level. First‑try answers score 2 points, later ones 1. Hints are there to use — the solution is a last resort.