OpenOlympiad
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Concept 19 of 19Foundation
Video

📌 Formula Sheet — Patterns

All the formulas from this chapter. Understand the WHY — don't just memorize.

Arithmetic nth term: aₙ = a + (n − 1)·d
Means: a = first term, d = common differenceUse when: constant addition
Why: You add d exactly (n−1) times to reach the nth term from the first.
⚠ Trap: It's (n−1), NOT n. First term hasn't added d yet.
Geometric nth term: aₙ = a · rⁿ⁻¹
Means: a = first term, r = common ratioUse when: constant multiplication
Why: You multiply by r exactly (n−1) times.
⚠ Trap: r can be less than 1 (halving, shrinking) — still geometric.
Counting terms: n = (last − first)/d + 1
Means: count items in an arithmetic listUse when: "how many multiples of X between A and B"
Why: (last − first) is the distance; d is each step; total steps + 1 (for starting term) = count.
⚠ Trap: Don't forget the "+1". Boundary mistakes are very common.
Sum of first n naturals: Sₙ = n(n+1)/2
Means: 1 + 2 + 3 + … + nUse when: adding consecutive naturals from 1
Why (Gauss): Pair first with last — each pair sums to (n+1), there are n/2 pairs → n(n+1)/2.
⚠ Trap: For sum from 5 to 20, do S(20) − S(4), not directly.
Sum of first n odd numbers: 1 + 3 + 5 + … + (2n−1) = n²
Means: odd numbers starting from 1Use when: summing consecutive odds
Why: Each added odd number extends the square-dot figure by one row+column, giving n².
⚠ Trap: Works only if you start from 1.
Sum of first n even numbers: 2 + 4 + … + 2n = n(n+1)
Means: evens starting from 2Use when: summing consecutive evens
Why: Factor 2 out: 2(1+2+…+n) = 2·n(n+1)/2 = n(n+1).
⚠ Trap: This is n(n+1), NOT n². Don't mix up with odd-sum formula.
Sum of squares: 1² + 2² + … + n² = n(n+1)(2n+1)/6
Means: squares of first n naturalsUse when: sum of squared terms
Why: Derived by induction or telescoping. Memorize the formula.
⚠ Trap: Don't confuse with (1+2+…+n)² — that's different!
Sum of cubes: 1³ + 2³ + … + n³ = [n(n+1)/2]²
Means: cubes of first n naturalsUse when: sum of cubed terms
Why (amazing!): Sum of cubes equals SQUARE of sum of naturals. 1+8+27+64 = 100 = (1+2+3+4)².
⚠ Trap: It's the square of the SUM of naturals, not the sum of squares.
Triangular number: Tₙ = n(n+1)/2
Means: dots in a triangle of n rowsUse when: stacking
Why: Same as sum of 1 to n, since you're adding row sizes 1, 2, 3, …, n.
⚠ Trap: Both Tₙ and squares can appear similar for small n — check the rule.
Tetrahedral (pyramid) number: Pₙ = n(n+1)(n+2)/6
Means: oranges in triangular-base pyramid of n layersUse when: stacking triangulars
Why: Pₙ = T₁ + T₂ + … + Tₙ. Summing triangular numbers using Σk² and Σk formulas gives n(n+1)(n+2)/6.
⚠ Trap: 1, 4, 10, 20, 35 looks like squares for small n — don't confuse.
Pentagonal: Pₙ = n(3n−1)/2; Hexagonal: Hₙ = n(2n−1)
Means: dots in pentagon / hexagon figureUse when: olympiad figurate problems
Why: Each added ring of the polygonal shape has (number of new dots) = specific arithmetic progression.
⚠ Trap: Pentagonal grows SLOWER than hexagonal. Don't mix.
Factorial: n! = 1·2·3·…·n, with 0! = 1
Means: product of 1 through nUse when: arrangements of n objects
Why: n people in a line: first slot has n choices, second (n−1), and so on. Multiply.
⚠ Trap: Factorials grow faster than any polynomial or exponential. 10! = 3,628,800.
Fibonacci: Fₙ = Fₙ₋₁ + Fₙ₋₂, F₁ = F₂ = 1
Means: each term = sum of previous twoUse when: recursive "add" patterns
Why: Named after Fibonacci's rabbit-breeding puzzle. Appears surprisingly in nature.
⚠ Trap: Start values (1, 1) matter. Different starts give a different sequence.
Repeating pattern nth item: use n mod (period)
Means: find which item in the blockUse when: A, B, C, D, A, B, C, D, …
Why: After every full block, pattern restarts, so only position WITHIN the block matters.
⚠ Trap: If n mod period = 0, use the LAST item of the block, not the first.
💡 Tip:When unsure, test your formula on small n (n=1, 2, 3) and match against known terms.
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