Practice Generalization in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
The process of extending a specific result or pattern to hold for a broader class of objects or situations.
Does this pattern work more generally? Can we remove restrictions?
Example 1
easyYou observe: 2+4=6, 4+6=10, 6+8=14. Formulate a general rule and prove it.
Example 2
mediumThe identity (a+b)^2 = a^2 + 2ab + b^2 is familiar. Generalise it to (a+b)^3 and state the pattern for (a+b)^n.
Example 3
easySpecific: 3 \times 5 = 15 (odd \times odd = odd). Generalise: prove that the product of any two odd integers is odd.
Example 4
mediumThe formula 1+2+\cdots+n = \frac{n(n+1)}{2} holds for n=1,2,3. State how you would generalise this claim to all positive integers and what technique would be used.