Technique Encyclopedia

Testing Small Cases

What it is

Testing small cases means computing a problem's quantity for the first few small inputs (n=1, 2, 3, ...) before attempting a general argument. On a multiple-choice test, the pattern you find is often enough to answer the question directly — you don't always need a full proof.

Signals that suggest using it

The problem involves a general n, a recursively defined sequence, or a repeated process.
A closed-form answer isn't obvious from the problem statement alone.
You suspect a cyclical pattern (units digits, remainders, parity) but haven't confirmed it.

When it's effective

This is extremely fast for finding cyclical patterns (units digits of powers, remainders under repeated operations) and for sanity-checking a formula you've derived by another method before committing to it as your final answer.

When it's not effective

A pattern that holds for the first several small cases isn't guaranteed to hold for the value the problem actually asks about — always confirm the cycle length or pattern's mechanism, not just extrapolate optimistically from a handful of terms.

Simple example

Problem

Find the units digit of 7^2025.

Solution

Compute the units digits of the first several powers of 7: 7^1=7, 7^2=49, 7^3=343, 7^4=2401, 7^5=16807 — the pattern 7, 9, 3, 1 repeats every 4 powers. Since 2025=4(506)+1, 7^2025 has the same units digit as 7^1: 7.

AMC-style example

Problem

A sequence satisfies a_1=1 and a_n = a_(n-1) + n for n ≥ 2. Find a_10.

Solution

Computing the first several terms: a_1=1, a_2=3, a_3=6, a_4=10 — these are the triangular numbers, a_n=n(n+1)/2. So a_10 = 10(11)/2 = 55, without computing all ten terms by hand.

Strategic Substitution Working Backwards