1.2 Proof by deduction — Practise

Work through these on your own. Begin each proof from the assumption, use a general letter throughout, and finish with a sentence in words. Write your working out in full: the reader should be able to follow every step.

Both the questions and the answers below come from a set that is checked by computer every time the book is built.

The sum of any two even numbers is even. Show that $2m + 2n$ is a multiple of $2$ .
Show that the sum of two odd numbers is even.
Show that the sum of two consecutive whole numbers is odd.
Show that $(n + 1)^2 - n^2 = 2n + 1$ .
Show that $n^2 + n$ is even for every whole number $n$ .
Write $x^2 + 6x + 11$ in the form $(x + a)^2 + b$ .
Show that $x^2 - 4x + 7$ is positive for every real number $x$ .
Show that $(3x + 1)(x - 2) = 3x^2 - 5x - 2$ .
Show that the product of two consecutive even numbers is a multiple of $4$ .
Show that the square of an odd number is odd.

Answers and concise solutions

1. Show the sum is a multiple of $2$

$2*m + 2*n$
$2*(m + n)$

Answer: Let the two even numbers be $2m$ and $2n$ . Their sum is $2m + 2n = 2(m + n)$ . The number $m + n$ is a whole number, so the sum has a factor $2$ : it is even.

2. Show the sum is even

Answer: Write the odd numbers as $2m + 1$ and $2n + 1$ . Their sum is $2m + 1 + 2n + 1 = 2m + 2n + 2 = 2(m + n + 1)$ . This is a multiple of $2$ , so the sum is even.

3. Show the sum is odd

Answer: Let the consecutive whole numbers be $n$ and $n + 1$ . Their sum is $2n + 1$ . This is one more than a multiple of $2$ , so it is odd.

4. Show the identity

Answer: Expand the bracket: $(n + 1)^2 - n^2 = n^2 + 2n + 1 - n^2$ . The $n^2$ terms cancel, leaving $2n + 1$ .

5. Show $n^2 + n$ is even

$n**2 + n$
$n*(n + 1)$

Answer: Factorise: $n^2 + n = n(n + 1)$ . These are two consecutive whole numbers. One of any two consecutive numbers is even, so the product has a factor $2$ and is even.

6. Complete the square

Answer: Half of $6$ is $3$ , and $(x + 3)^2 = x^2 + 6x + 9$ . So $x^2 + 6x + 11 = (x + 3)^2 + 2$ .

7. Show the expression is always positive

Answer: Complete the square: $x^2 - 4x + 7 = (x - 2)^2 + 3$ . A square is never negative, so $(x - 2)^2 \geq 0$ . Adding $3$ gives at least $3$ , which is positive.

8. Show the identity

Answer: Expand: $(3x + 1)(x - 2) = 3x^2 - 6x + x - 2 = 3x^2 - 5x - 2$ .

9. Show the product is a multiple of $4$

Answer: Write the consecutive even numbers as $2m$ and $2m + 2$ . Their product is $4m(m + 1)$ . This has a factor $4$ , so it is a multiple of $4$ .

10. Show the square is odd

Answer: Write the odd number as $2n + 1$ . Its square is $(2n + 1)^2 = 4n^2 + 4n + 1 = 2(2n^2 + 2n) + 1$ . This is one more than a multiple of $2$ , so it is odd.