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MathidiaMathematics Academy
Mathematical research methods
Lesson 67 of 75
Easy practice

Conjecture and experimentation

Objective. Use examples and computation to form testable conjectures.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Conjecture and experimentation uses specific examples and computation to notice a pattern, state it as a precise conjecture, and then test that conjecture against further cases before attempting a proof. For example, computing the sum of the first n odd numbers for n = 1, 2, 3, 4 gives 1, 4, 9, 16, suggesting the conjecture that the sum equals n². Testing n = 5 gives 1+3+5+7+9 = 25 = 5², supporting the conjecture, which can then be proved by induction. Good experimentation also tries to find a counterexample deliberately, since a conjecture that survives serious attempts to break it is more trustworthy before a proof is found.

Worked example

In forming a conjecture by experimentation, compute the sum of the first n cubes for n = 1, 2, 3 (values 1, 9, 36) and compare with (n(n+1)/2)². What value does the conjectured formula give for n = 4?

Diagram of a cube
  1. 1Check the pattern: for n=1,2,3 the sums 1, 9, 36 match (n(n+1)/2)² exactly.
  2. 2Apply the conjectured formula with n = 4: n(n+1)/2 = 4(5)/2 = 10.
  3. 3Square this result: 10² = 100.
  4. 4So the conjectured sum of the first 4 cubes is 100, matching 1+8+27+64.

Answer: 100

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Common mistake. A common mistake is treating a pattern confirmed by only a few small cases as proven, such as believing n²+n+41 is always prime because it is prime for small n, when in fact it fails at n = 40.

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