Special & Boundary Cases

Test the extremes first

ParametersIntuitionTesting

Before attempting the general case, try the extremes. A special case occurs when a variable takes an extreme value: $n=1$, $x=0$, $k=n$, etc. These cases often reveal whether a claim is true and how to prove it.

When to use it

→When the problem has a parameter (e.g., $n$, $k$)
→When a proof is required for 'every $n$'
→When you don't know where to start
→When the general case seems complex

How to think (step by step)

1Identify the parameter: Which variable takes different values?
2Try $n = 1$: What happens in the simplest case?
3Try $n = 2$: And the next?
4Look for a pattern: What do you observe?
5Generalize: Now try to prove it for all $n$

Practice Problems

Three problems at increasing difficulty — try each before revealing the hint or solution.

📘Basic

A student claims that for every positive integer

\(n\)

, the number

\(n^2 + n + 41\)

is prime. Determine whether this is true or false.

📙Intermediate

Prove that for every positive integer

\(n\)

, the number

\(n^3 - n\)

is divisible by

\(6\)

📕Advanced

Find all functions

f: \mathbb{R} \to \mathbb{R}

satisfying

\(f(x^2 + f(y)) = y + f(x)^2\)

for all

x, y \in \mathbb{R}