Algebraic Proof with Odd, Even and Consecutive Numbers
Prove a general claim about integers by naming them algebraically, expanding, and showing the required factor is always there.
What a learner can do afterwards
- Write any even number, any odd number and consecutive integers in terms of n
- Prove that the product of two consecutive integers is even, showing the factor explicitly
- Say why an algebraic argument settles a claim that no number of tested cases can
1 · Read
Proofs start by writing numbers in general form. Any even number is 2n, any odd number is 2n plus 1, and consecutive integers are n and n plus 1. These definitions turn wordy claims into algebra you can push around.
To prove the product of two consecutive integers is even, write n times n plus 1. Neighbours always split odd and even, so one factor is even. Pull the 2 out front, as in 7 times 8 equals 2 times 28.
Sums work the same way. Two consecutive integers add to n plus n plus 1, which is 2n plus 1, the odd form. Equivalent expressions like 2n plus 2 and 2 times n plus 1 match for every n, so swapping them is always safe.
No count of tested cases proves a claim for all integers, since cases never cover infinity. Algebra over n settles every case at once. Try it: three consecutive integers sum to 3n plus 3, which is 3 times n plus 1, always a multiple of three.
Name numbers with n, display the factor the claim needs, and let the algebra cover every integer at once.
More to read
A | Proofs, Identities, and Toolkit Functions
Proofs start by writing numbers in general form: any even number is 2n, any odd number is 2n + 1, and consecutive integers are n and n + 1. These definitions turn wordy claims into algebra you can push around.
To prove the product of two consecutive integers is even, write n(n + 1) and note one factor is always even. Pull out the factor 2 explicitly to finish. The algebra covers every integer at once, which testing never can.
How to Add Consecutive Integers from 1 to 100: 9 Steps
Pairing consecutive numbers reveals structure: 1 + 100, 2 + 99, and so on, each pair totalling 101. Fifty such pairs give 5050. The pairing argument works for any similar run, and algebra generalises it fully.
Proofs use the same instinct one level up. Instead of pairing examples, write the general pair n and n + 1 and work with symbols.
6.4: Writing Equivalent Expressions
Equivalent expressions are the engine of proof: 2n + 2 and 2(n + 1) match for every n, so swapping them is always safe. Rewriting sums of consecutive integers into 2n + 1 form shows oddness in one glance.
Practise rewriting until the even or odd factor jumps out. If you can display a factor 2, the number is even; a remainder 1 after factoring 2 means odd.
1.5 Factoring Polynomials
Showing a factor explicitly is what makes divisibility proofs stick. Writing 7 times 8 as 2 times 28 displays the 2 that guarantees evenness. Bare answers like 56 hide the reason.
Factor first, conclude second. For n(n + 1), point at the even factor in the pair, pull it out front, and the product is visibly 2 times something.
2 · Watch
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Where it sits
Optional connections
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Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.