Generalization Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Specific: 3×5=153 \times 5 = 15 (odd ×\times odd = odd). Generalise: prove that the product of any two odd integers is odd.

Solution

  1. 1
    Let the two odd integers be 2m+12m+1 and 2n+12n+1.
  2. 2
    Product: (2m+1)(2n+1)=4mn+2m+2n+1=2(2mn+m+n)+1(2m+1)(2n+1) = 4mn+2m+2n+1 = 2(2mn+m+n)+1.
  3. 3
    This is of the form 2k+12k+1 (where k=2mn+m+nk=2mn+m+n), so it is odd.

Answer

(2m+1)(2n+1)=2(2mn+m+n)+1 is odd(2m+1)(2n+1) = 2(2mn+m+n)+1 \text{ is odd}
Starting from a specific example and replacing specific numbers with variables is the fundamental move in generalisation. The algebraic proof covers all cases at once.

About Generalization

The process of extending a specific result or pattern to hold for a broader class of objects or situations.

Learn more about Generalization →

More Generalization Examples

Example 1 easy

You observe: [formula], [formula], [formula]. Formulate a general rule and prove it.

Example 2 medium

The identity [formula] is familiar. Generalise it to [formula] and state the pattern for [formula].

Example 4 medium

The formula [formula] holds for [formula]. State how you would generalise this claim to all positive

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Related Concepts

AbstractionSpecialization