Write whole numbers in general (n, 2n, 2n + 1), expand and simplify, and conclude for every case at once.
What you will learn
- Write even, odd and consecutive numbers with algebra
- Show an identity by expanding
- Prove a result about every whole number
General numbers
Checking examples is not a proof: the result must hold for every whole number.
Use letters.
- An even number: 2n. An odd number: 2n + 1.
- Consecutive numbers: n, n + 1, n + 2. Consecutive even numbers: 2n, 2n + 2.
- A multiple of 3: 3n.
An expression is always even when 2 is a factor: 2n + 4 = 2(n + 2). In the same way, 3n + 6 = 3(n + 2) is always a multiple of 3, and 2n + 1 is always odd. But n + 1 can be odd or even, depending on n.
Two different odd numbers need two letters: 2a + 1 and 2b + 1.
The rest of this lesson and its 7 practice questions are locked.
In this lesson
- General numbers
- Expand, simplify, conclude
- Identities
- Odd, even and consecutive
7 practice questions, marked as you go; about 25 minutes in all.
Key terms
Identity
In this unit: Proof and iteration
Algebraic proof and counterexamples, then finding roots that do not come out exactly: changes of sign and iteration. Higher tier.
- Algebraic proof
- Counterexamples
- Roots by a change of sign
- Iteration