Pass Check: Algebraic Proof
Use algebra to show a general statement is always true, using n for any integer.
Proof
A logical argument using algebra that shows a statement is true for all cases, not just examples.
Consecutive integers
Whole numbers that follow each other in order, written as n, n+1, n+2, ...
Even and odd numbers
An even number can be written as 2n; an odd number can be written as 2n + 1, for any integer n.
Identity
A statement that is true for all values of the variable, shown using the ≡ symbol.
Quick check
Mixed quiz
1. Prove that the product of two consecutive integers n(n+1) is always even. Which factor shows this?
2. If n represents any integer, what does 2n represent?
3. Why does 8n²+16n+10 show the sum is even?
4. Why does 4n²+4n+1 prove (2n+1)² is odd?
5. Which counter-example disproves: 'the square of any number is positive'?
6. For x³-x-3=0, if f(1.65)=0.106 (positive), what does this tell us about the root to 2dp, given f(1.5)<0?
7. An iteration formula gives x₃ = 3.162, x₄ = 3.162 (to 3 dp). What can you conclude?
8. The equation x³ + x - 5 = 0 can be rearranged to x = (5-x)^(1/3)... which formula below correctly rearranges x³ = 5 - x for iteration?