Proof by deduction

Direct proof — starting from things known to be true and reasoning in valid algebraic steps to the conclusion. Representing the general case (2n, 2n+1, consecutive integers), "show that" manipulations, completing the square for positivity, and always finishing with a clear conclusion.

Pure · Proof

Proof by deduction

Why it works

A proof by deduction (a direct proof) starts from statements already known to be true — definitions and established results — and moves by valid steps to the thing you must show. Done properly it settles the claim for every case at once, which is exactly why checking a few examples is not a proof.

The single most important habit is to represent the general case with algebra, not with particular numbers:

any integer: $n$ ;
an even number: $2n$ ; an odd number: $2n + 1$ ;
consecutive integers: $n$ and $n + 1$ ;
consecutive even numbers: $2n$ and $2n + 2$ ; consecutive odd: $2n+1$ and $2n+3$ ;
a multiple of 3: $3n$ .

If a statement is about "all odd numbers", you must start "let

n = 2k + 1

…", work in that letter, and reach the conclusion — never "well,

3

works and

5

works".

For a "show that" question, the work is to manipulate the expression into a form that makes the required property obvious:

to show something is even, write it as $2 \times (\text{integer})$ ;
to show it is a multiple of 4, factor out a $4$ ;
to show it is always positive, complete the square: anything of the form

(x - a)^2 + b

with

b > 0

is positive for every real

x

, because a square is never negative.

Finish with a concluding statement that links back to the claim ("…which is a multiple of 4, as required"). Examiners award the final mark for that conclusion.