---
title: "Algebraic Proof with Odd, Even and Consecutive Numbers"
description: "Prove a general claim about integers by naming them algebraically, expanding, and showing the required factor is always there."
canonical: https://lightmysky.com/learn/mathematics/algebraic-proof-with-odd-even-and-consecutive-numbers-mt_p9bajOC17q
source: https://lightmysky.com/learn/mathematics/algebraic-proof-with-odd-even-and-consecutive-numbers-mt_p9bajOC17q.md
retrieved: 2026-09-12
---

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# 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.

Subject: Mathematics · Area: Mathematical Thinking · Ages 15 to 16
Page: https://lightmysky.com/learn/mathematics/algebraic-proof-with-odd-even-and-consecutive-numbers-mt_p9bajOC17q

## Ready when they can

- 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

## Lesson: Proving with odd, even, and consecutive

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.

**Example.** 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. That visible 2 is the whole proof.

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.

**Tip.** 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.

**Recap.** Name numbers with n, display the factor the claim needs, and let the algebra cover every integer at once.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [Difference of Two Squares and Perfect Square Trinomials](https://lightmysky.com/learn/mathematics/difference-of-two-squares-and-perfect-square-trinomials-mt_GSpsHuserT)
- [Expanding Double Brackets](https://lightmysky.com/learn/mathematics/expanding-double-brackets-mt_KhS7K1Mgrw)
- [Modelling Situations with Formulae and Graphs](https://lightmysky.com/learn/mathematics/modelling-situations-with-formulae-and-graphs-mt_z5iwdZyeDr)

## Opens up

- [Proof by Deduction and Exhaustion](https://lightmysky.com/learn/mathematics/proof-by-deduction-and-exhaustion-mt_QNWvbkg04f)