AeroLab Reading room

MATH / A2 level

Proof, modelling and component choices

A proof establishes a statement from assumptions using valid deductions; examples can suggest a conjecture but cannot prove a universal claim. A counterexample disproves one.

Learning objectives

The big idea

A proof establishes a statement from assumptions using valid deductions; examples can suggest a conjecture but cannot prove a universal claim. A counterexample disproves one.

Cambridge 9709 offers component combinations rather than one list studied by every candidate. Check the permitted pure, mechanics and statistics route; model assumptions and communication matter in every option.

Equations and relationships

n2+n=n(n+1)n^2+n=n(n+1)

The product of consecutive integers is even.

Worked example

Why is n²+n even for every integer n?

Follow the solution

Factor as n(n+1). Consecutive integers contain one even factor, making the product even.

Investigate and explain

Predict the worked result before calculating. Change one input, record the new result, and explain which assumptions still hold.

Use the prediction activity below; a local model appears only where it supports this topic.

Open the interactive investigation

Check your understanding

1. What disproves a statement claiming every integer has a property?

  1. One supporting example
  2. One valid counterexample
  3. A rounded graph
Check the answer and reasoning

One valid counterexample

A universal statement fails if a single allowed case contradicts it.

2. Why is n²+n even for every integer n?

  1. Both factors are odd
  2. It was checked for n=1
  3. One of n and n+1 is even
Check the answer and reasoning

One of n and n+1 is even

Factor as n(n+1). Consecutive integers contain one even factor, making the product even.

3. May every student assume all component options are compulsory?

  1. No
  2. Yes
  3. Only for statistics
Check the answer and reasoning

No

The syllabus specifies permitted routes; component choices must match the examination entry.