AQA Level 2 Further Maths · Level 2 Certificate · Chapter 4: Algebra IV · Lesson 6

Algebraic proof

About 80 minutesNo calculator70 marks

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Date

Do Now

The first two questions are what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Factorise fully $5 x^{2} - 45$ .
[2 marks]
Answer
A2
Needed today
Write $x^{2} + 8 x + 27$ in the form $(x + a)^{2} + b$ .
[2 marks]
Answer $(x + \dots\dots)^{2} + \dots\dots$
A3
Last lesson
Solve $8^{x} = 32$ .
[2 marks]
Answer $x = \dots\dots$
A4
From chapter 4
Solve $x^{2} - x - 12 < 0$ .
[2 marks]
Answer

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it.

Example 1

Prove that $4 (2 n + 9) - 2 (n - 3)$ is a multiple of $6$ for every integer $n$ .

Hint Expand both brackets and collect like terms. Then take out the common factor.

B1
Prove that $6 (2 n + 5) - 3 (n - 2)$ is a multiple of $9$ for every integer $n$ .
[3 marks]

Example 2

Prove that $4 n^{2} (16 n - 3) + 12 n^{2}$ is a cube number when $n$ is a positive integer.

Hint Expand the bracket and simplify. Then write what is left as something cubed.

B2
Prove that $3 k (12 k + 5) - 15 k$ is a square number when $k$ is a positive integer.
[3 marks]

Example 3

Write $x^{2} - 12 x + 41$ in the form $(x - a)^{2} + b$ . Hence prove that $x^{2} - 12 x + 41 > 0$ for all values of $x$ .

Hint Halve the

12

to get the number in the bracket, then take off

6^{2}

. A square is never negative.

B3
Prove that $x^{2} + 14 x + 51 > 0$ for all values of $x$ .
[4 marks]

Example 4

$a$ and $b$ are positive integers with $a > b$ . Prove that $\frac{a x ^{2} + 5 a x}{b x ^{2} + 5 b x} > 1$ , where $x \neq = 0$ and $x \neq = - 5$ .

Hint Factorise the top and the bottom. Look for the factors they share.

B4
Prove that $\frac{12 - 4 y}{y ^{2} - 9}$ is negative for every $y > - 3$ , where $y \neq = 3$ .
[4 marks]

Example 5

$f (n) = n^{2} + 3 n$ . Prove that $f (n + 1) - f (n - 1)$ is always even.

Hint Put

n + 1

, then

n - 1

, in place of

n

. Expand, subtract, and take out a factor of

2

B5
$f (x) = x^{2} + 4 x$ . Prove that $f (3 x) - f (x) = k x (x + 1)$ , where $k$ is an integer.
[3 marks]

Practice

Level 2

C1
Prove that $3 (4 n - 1) - 4 (3 n - 5)$ is the same positive integer for every value of $n$ .
[2 marks]
C2
Prove that the product of two consecutive odd numbers is one less than a multiple of $4$ .
[3 marks]

C3
$f (x) = (2 x - 5)^{2} + 8 (2 x - 3)$ . Prove that there is exactly one value of $x$ for which $f (x) = 0$ .
[3 marks]
C4
Prove that $10 x - x^{2} - 30$ is negative for all values of $x$ .
[3 marks]
C5
Prove that $\frac{15 p - 20}{3 p - 4}$ is always a positive integer, where $p \neq = \frac{4}{3}$ .
[2 marks]
C6
$p$ is a positive number and $q$ is a negative number. Prove that $\frac{3 p ^{2} - 3 pq}{pq - q ^{2}}$ is negative.
[3 marks]

Exam-style questions

AQA exam style

D1
Priya says, " $(n + 3)^{2} - n^{2}$ is a multiple of $6$ for every positive integer $n$ ."
$n$ is a positive integer.
(a)
Prove that $(n + 3)^{2} - n^{2}$ is a multiple of $3$ .
[3 marks]
(b)
Explain why Priya is wrong.
[1 mark]
Total for D1: 4 marks

D2
The $n$ th term of a sequence is $n^{2} + 2 n$ .
(a)
Work out an expression for the $(n + 1)$ th term of the sequence. Give your answer in its simplest form.
[2 marks]
Answer $\dots\dots\dots\dots$
(b)
Hence prove that the difference between any two consecutive terms of the sequence is an odd number.
[2 marks]
(c)
Prove that one more than any term of the sequence is a square number.
[2 marks]
Total for D2: 6 marks
D3
$f (x) = 2 x^{2} - 16 x + 37$
(a)
Write $f (x)$ in the form $a (x + b)^{2} + c$ , where $a$ , $b$ and $c$ are integers.
[3 marks]
Answer $\dots\dots (x + \dots\dots)^{2} + \dots\dots$
(b)
Hence prove that $f (x) > 0$ for all values of $x$ .
[1 mark]
(c)
Hence work out the greatest value of $\frac{10}{f ( x )}$ .
[2 marks]
Answer $\dots\dots$
Total for D3: 6 marks
D4
Prove that $\frac{2 x ^{2} + 6 x}{x ^{2} - 9} \div \frac{x ^{2} + x}{x ^{2} - 2 x - 3} = 2$ for every value of $x$ for which it is defined. You must show your working.
[4 marks]
D5
$g (n) = n^{2} - 2 n$
(a)
Show that $g (n + 1) = n^{2} - 1$ .
[2 marks]
(b)
Hence prove that $g (n + 1)$ is a multiple of $8$ when $n$ is an odd positive integer.
[3 marks]
Total for D5: 5 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Prove that the difference between the cubes of two consecutive odd numbers is always $2$ more than a multiple of $24$ .
[4 marks]
Hint 1 · What to try
Write the two odd numbers as $2 n - 1$ and $2 n + 1$ , and expand each cube.
Hint 2 · The first line
$(2 n + 1)^{3} = 8 n^{3} + 12 n^{2} + 6 n + 1$ and $(2 n - 1)^{3} = 8 n^{3} - 12 n^{2} + 6 n - 1$ .
Hint 3 · The full method
Subtract the second cube from the first: the $8 n^{3}$ and $6 n$ terms cancel, leaving $24 n^{2} + 2$ . That is $2$ more than $24 n^{2}$ , a multiple of $24$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can prove a result about integers by writing an expression as a multiple, a square or a cube.
I can complete the square to prove an expression is always positive or always negative.
I can factorise and cancel algebraic fractions to prove a result.
I can use function notation and forms such as $2 n + 1$ in a proof.

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