courses › Counting & Discrete Math

Sequences defined step by step, and their shortcuts.

Pen and paper is fine · no calculator needed why?

Learn first (about 3 minutes)

Opens at level 42.

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Builds on: Sequences & series (not open yet)

skills · practice stats

From rounds of this course only: box, review and trial answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

• Fibonacci par 18 s not tried yet

  worked example

  F₁ = F₂ = 1, and each term is the sum of the two before it. What is F₉?

  Answer: 34

  1. Keep adding the last two terms: 1, 1, 2, 3, 5, 8, 13, 21, 34.
  2. So F₉ = 34.

• Step through a rule par 20 s not tried yet

  worked example

  a₁ = 2 and aₙ = 3aₙ₋₁ − 1. What is a₅?

  Answer: 122

  1. Apply the rule one step at a time: a₂ = 3 × 2 − 1 = 5.
  2. Then a₃ = 14, a₄ = 41, a₅ = 122.

• Tower of Hanoi par 17 s not tried yet

  worked example

  What is the minimum number of moves to solve the Tower of Hanoi with 8 disks?

  Answer: 255

  1. Moving n disks means moving n − 1 disks aside, the big one once, then the n − 1 back on top: the count doubles and adds one each time, giving 2ⁿ − 1.
  2. 2⁸ − 1 = 256 − 1 = 255.

• Spot the formula par 30 s not tried yet

  worked example

  a₀ = 0 and aₙ = aₙ₋₁ + 2n − 1. What is a₂₄?

  Answer: 576

  1. The terms go 0, 1, 4, 9, 16, …: adding the odd numbers 1, 3, 5, 7, … gives the squares, so aₙ = n².
  2. a₂₄ = 24² = 576.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1. 1 day
2. 3 days
3. 7 days
4. 14 days
5. 30 days
6. 60 days
7. mastered · every 90 days

your rounds

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