What is proof by deduction?

Using logical steps and algebra from known facts to show a result is true for all cases.

What is proof by exhaustion?

Splitting into a finite number of cases and proving the statement in each case.

Why use 2m and 2n for two even numbers?

They are independent integers. One letter would force the two numbers to be equal.

Show n^2+n is always even.

n^2+n=n(n+1), a product of consecutive integers, one of which is even.

Does checking many examples prove a statement?

No. It supports the statement; only a general argument proves it.

Cases for division by 3?

Every integer is 3k, 3k+1 or 3k+2.

How do you show an integer is odd?

Write it as 2(\ldots)+1 with an integer bracket.

What must the end of a proof do?

State the result clearly and link it back to the question.

Proof by deduction and exhaustionAQA A-Level Maths: Flashcards

Card 1 of 130 of 13 known

Question

How do you write any even integer and any odd integer algebraically?

Tap or press Space to reveal

Tap card or press Space to flip

See all 13 cards

How do you write any even integer and any odd integer algebraically?: Even: $2n$ . Odd: $2n+1$ , where $n$ is an integer.
What is proof by deduction?: Using logical steps and algebra from known facts to show a result is true for all cases.
What is proof by exhaustion?: Splitting into a finite number of cases and proving the statement in each case.
Algebraic form of three consecutive integers?: $n$ , $n+1$ , $n+2$
Algebraic form of two consecutive odd numbers?: $2n-1$ and $2n+1$
Why use $2m$ and $2n$ for two even numbers?: They are independent integers. One letter would force the two numbers to be equal.
Show $n^2+n$ is always even.: $n^2+n=n(n+1)$ , a product of consecutive integers, one of which is even.
Does checking many examples prove a statement?: No. It supports the statement; only a general argument proves it.
Cases for division by 3?: Every integer is $3k$ , $3k+1$ or $3k+2$ .
Factorise $n^3-n$ .: $(n-1)n(n+1)$
Which digits can the last digit of a square number be?: $0,1,4,5,6,9$
How do you show an integer is odd?: Write it as $2(\ldots)+1$ with an integer bracket.
What must the end of a proof do?: State the result clearly and link it back to the question.

Exam questions on Proof by deduction and exhaustion

Let $n$ be a positive integer.
Show that the sum of the three consecutive integers $n$ , $n+1$ and $n+2$ is a multiple of 3.2 marks
Every integer has the form $3k$ , $3k+1$ or $3k+2$ , where $k$ is an integer.
Hence prove that the square of any integer is either a multiple of 3 or one more than a multiple of 3.2 marks
A student wants to prove that $n^3-n$ is divisible by 6 for every positive integer $n$ .
Factorise $n^3-n$ fully and use your factorisation to explain why it is always even.3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).