questions

Proof by Deduction &amp; Exhaustion

Let <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>n be an integer.
Prove that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">n(n+1)</annotation></semantics></math>n(n+1)
is even.

Recognises that one of the consecutive integers <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>n and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>n+1 must be even.

Concludes that <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">n(n+1)</annotation></semantics></math>n(n+1) is even because its factors include an even integer.

Proof by Deduction & Exhaustion Questions

Proof

Proof by Deduction & Exhaustion

Algebra & Functions

Indices & Surds

Quadratics

Simultaneous Equations

Inequalities

Polynomials

Graphs & Transformations

Modelling with Functions

Coordinate Geometry

Straight Lines

Circles

Sequences & Series

Binomial Expansion

Trigonometry

Trigonometric Ratios & Rules

Trigonometric Graphs

Trigonometric Identities & Equations

Exponentials & Logarithms

Exponential Equations

Laws of Logarithms

Modelling with Exponentials & Logs

Differentiation

Differentiation from First Principles

Tangents & Normals

Stationary Points

Integration

Definite Integration

Vectors

Two-Dimensional Vectors