AMC 8 Interactive Masterclass: Sequences, Combinatorics, Functions, Binomials, Polynomials, Probability

Topic 1

Sequences

Spot the pattern fast: constant difference (arithmetic), constant ratio (geometric), constant second difference (quadratic), or linear recurrences. Verify with early terms before committing.

Models

\ \[a_n = a_1 + (n-1)\\cdot d\\quad\\text{(arithmetic)}\\]

\ \[a_n = a_1\\cdot r^{\\,n-1}\\quad\\text{(geometric)}\\]

\ \[a_n = An^2 + Bn + C\\quad\\text{(quadratic trend)}\\]

Quick check

Identify: Sequence: 2, 7, 14, 23, … What model fits best?

Hint: check first differences, then second differences.

Compute

Arithmetic: If \(a_1=4\) and \(d=3\), compute \(a_8\).

Mini-game: difference detector

Toggle tiles to reveal differences and decide the type.

Revealed: 0/5

Topic 2

Stars and bars

Count nonnegative or positive integer solutions. Convert “distribute n identical items to k boxes” to separators and choose positions.

Formulas

\ \[x_1+\\cdots+x_k=n,\\;x_i\\ge0\\;\\Rightarrow\\;\\binom{n+k-1}{k-1}\\]

\ \[x_1+\\cdots+x_k=n,\\;x_i\\ge1\\;\\Rightarrow\\;\\binom{n-1}{k-1}\\]

Choose count

Nonnegative: Solutions to \(x_1+x_2+x_3=20\) with \(x_i\ge0\) equals?

Hint: n+k-1 choose k-1.

Bounded variant

Bounded: Number of solutions to \(a+b+c=10\) with \(0\le a\le5\), \(b,c\ge0\).

Enter a count; consider subtracting cases with \(a\ge6\).

Hint: inclusion–exclusion on a.

Mini-game: bar builder

Place bars among stars to visualize \\(x_1+x_2+x_3=n\\). Click to toggle bars.

n: Arrangements: 0

Topic 3

Functions and operations

Apply definitions precisely. For custom operations, rewrite and simplify carefully. For compositions, evaluate inner first, check domain constraints.

Compose

If \(f(x)=2x-1\) and \(g(x)=x+4\), compute \(f(g(7))\) and \(g(f(7))\).

Inverse

Let \(h(x)=\\frac{x-5}{2}\\). Find \(h^{-1}(y)\).

Mini-game: operation lab

Custom op: \(a \\blacktriangle b = 2a+b^2\). Build expressions and evaluate.

Result: —

Nested: —

Topic 4

Binomials and Pascal’s triangle

Coefficients in \((a+b)^n\) are binomial coefficients. Use symmetry and the Pascal recurrence for quick checks.

\ \[(a+b)^n = \\sum_{k=0}^{n} \\binom{n}{k} a^{n-k} b^{k}\\]

Coefficient finder

Find coefficient of \(x^4\) in \((x+3)^7\).

Hint: choose k such that \(k=4\).

Sum identity

Evaluate \(\\sum_{k=0}^5 \\binom{5}{k}\\).

Hint: \((1+1)^5\).

Mini-game: Pascal painter

Generate rows and highlight a coefficient.

Rows:

Topic 5

Polynomials

Factor efficiently, use remainder theorem, and exploit root relations. Many word problems encode simple polynomial structure.

Factor & roots

Factor \(x^2-7x+12\) and give its roots.

Remainder theorem

If \(P(x)=3x^3-2x+5\), remainder when dividing by \(x-2\)?

Mini-game: synthetic sprint

Test candidate roots quickly; if \(P(a)=0\), factor out \((x-a)\).

P(a): —

Topic 6

Probability

Translate to counts: favorable over total. Use addition for exclusive events, multiplication for independent events, and complement for “at least” patterns.

Dice

Two fair dice: probability the sum is 7?

Complement

Draw two cards without replacement from a 52-card deck. Probability of at least one ace?

Hint: 1 − P(no ace on both draws).

Mini-game: coin combos

Three fair coins tossed. Toggle outcomes you think count for “exactly two heads,” then check.

Your count: 0 / Correct: 3

Synthesis

Mixed challenge set

Answer and track streaks. Challenges combine multiple topics the way AMC often does.

C1

Sequence + recurrence: \(a_n=2a_{n-1}-a_{n-2}\) with \(a_1=2, a_2=5\). Find \(a_6\).

Hint: characteristic equation.

C2

Stars + probability: Distribute 5 identical treats to 3 kids. Pick a random kid; probability they received at least 2?

Hint: count distributions; symmetry.

C3

Binomial identity: Compute \(\\sum_{k=0}^{5} \\binom{5}{k}2^{k}\\).

Hint: binomial theorem with \(a=1,b=2\).

C4

Functions: \(f(x)=x^2-4x+7\). Find \(f(f(1))\).

C5

Polynomial roots: \(x^2-5x+6=0\). Compute \(\\frac{1}{r} + \\frac{1}{s}\) for roots \(r,s\).

Hint: use Vieta and simplify.

C6

Probability + combinations: 4 red, 3 blue, 3 green marbles; draw 2 without replacement. Probability same color?

Hint: count same-color pairs over total \\(\\binom{10}{2}\\).

Streak: 0

Drills

Speed drills

One-minute sprint: answer as many as you can. Your score and accuracy update live.

00:00 Score: 0 | Attempts: 0

Press Start to begin.