 # Use Formal Algebraic Proof

In this worksheet, students will learn how to express mathematical proofs and the features of formal algebraic proof which students need to be familiar with using. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra

Curriculum subtopic:   Notation, Vocabulary and Manipulation, Algebraic Expressions

Difficulty level:   ### QUESTION 1 of 10

We can use algebraic proof to show the truth of many mathematical statements.

e.g. Prove that (n + 4)2 - (n + 2)2 is always a multiple of 4.

Start by expanding the brackets then simplifying.

You can use whichever method you prefer to expand the brackets, such as FOIL or the grid method.

Remember that when we see a square (2) on a bracket, we need to multiply the whole thing by itself.

(n + 4)2 using Grid Method

 n +4 n n2 +4n +4 +4n +16

= n2 + 4n + 4n + 6 = n2 + 8n + 6

(n + 2)2 using FOIL Method

First: n × n = n2

Outer: n × 2 = 2n

Inner: 2 × n = 2n

Last: 2 × 2 = 4

= n2 + 2n + 2n + 4 = n2 + 4n + 4

n2 + 4n + 4n +16 - (n2 +2n + 2n + 4)

= n2 + 4n + 4n + 16 - n2 - 2n - 2n - 4

= 4n + 12

Then we need to factorise to show that 4 is a factor and that the expression is a multiple of 4:

4(n + 3)

We use a set of expression to show different kinds of numbers.

These help us with writing algebraic proofs:

 2n an even number 2n + 1 an odd number n, n+1, n+2 consecutive numbers 2n, 2n+2, 2n+4 consecutive even numbers 2n+1, 2n+3, 2n+5 consecutive odd numbers 6n multiple of 6 5n multiple of 5

Have a look at this question now.

e.g. Prove that the sum of any three consecutive even numbers is always a multiple of 6.

We are going to show that adding three consecutive even numbers (2n, 2n+2, 2n+4) gives a multiple of 6:

2n + 2n + 2 + 2n + 4

Simplify:

6n + 6

Factorise to show 6 is a factor:

6(n + 1)

In this activity, we will learn more about how to express mathematical proofs and the features of formal algebraic proof which you need to be familiar with using.

You may want to have a pen and paper handy to record your working, as some of these proofs can become quite complex.

Match each expression below to its correct description.

## Column B

an odd number
2n+1, 2n+3, 2n+5, etc.
a multiple of 7
2n, 2n+2, 2n+4, etc.
consecutive even numbers
2n + 1
any number
n
consecutive odd numbers
2n
an even number
7n

Pick the term below which means the same as multiplying.

Sum

Difference

Product

Quotient

Select the correct label for each of the expressions below.

 Odd number Multiple of 3 Consecutive numbers Even number 3n n; n+1; n+2 2n+1 2n

Prove that (n + 10)2 - (n + 2)2 is a multiple of 16 for all positive values of n.

Write your full proof on a separate piece of paper then record your final concluding statement below.

Do not use a space between any of the numbers or signs in your final expression, or you may be marked incorrectly.

Prove that (4n + 1)2 - (4n - 1)2 is a multiple of 8 for all positive values of n.

Write your full proof on a separate piece of paper then record your final concluding statement below.

Do not use a space between any of the numbers or signs in your final expression, or you may be marked incorrectly.

Prove that (2n + 1)(n + 3) + (2n + 1)(n - 2) is an odd number.

Write your full proof on a separate piece of paper then record your final concluding statement by choosing from the options below.

4n² + 2n + 1

4(2n² + n) + 3

2(2n² + n) + 1

6n² + 3n + 1

Show algebraically that the product of two even numbers is always a multiple of 4.

Write your full proof on a separate piece of paper then record your final concluding statement by choosing from the options below.

4(n2)

n2 + 4

4(n - 1)

4(n + 1)2

Prove algebraically that the sum of two consecutive whole numbers is always odd.

Write your full proof on a separate piece of paper then record your final concluding statement by choosing from the options below.

2n

2n + 1

n + 2

3n

Prove algebraically that the difference between the squares of two consecutive odd numbers is always even.

Write your full proof on a separate piece of paper then record your final concluding statement below.

Do not use a space between any of the numbers or signs in your final expression, or you may be marked incorrectly.

True or false?

12n always represents an even number.

True

False

• Question 1

Match each expression below to its correct description.

## Column B

an odd number
2n + 1
a multiple of 7
7n
consecutive even numbers
2n, 2n+2, 2n+4, etc.
any number
n
consecutive odd numbers
2n+1, 2n+3, 2n+5, etc.
an even number
2n
EDDIE SAYS
In mathematical proof, n represents any number. If you want to represent an even number, it must be 2n, whereas an odd number will be 2n+1 (i.e. 1 more than an even number). Have a look at the table in the Introduction again if you need to remind yourself of any of these descriptions before moving on to the rest of this activity.
• Question 2

Pick the term below which means the same as multiplying.

Product
EDDIE SAYS
'Product' means the same as 'multiply'. e.g. the product of 3 and 4 is 12. 'Sum' means adding, 'difference' means subtracting and 'quotient' means dividing. Try to commit these terms to memory so that you can recognise them immediately in questions.
• Question 3

Select the correct label for each of the expressions below.

 Odd number Multiple of 3 Consecutive numbers Even number 3n n; n+1; n+2 2n+1 2n
EDDIE SAYS
3n denotes a multiple of 3. n, n+1, n+2 represent consecutive numbers. 2n+1 is an odd number, while 2n is an even number. Make sure you remember these. They will come in handy in the questions that follow.
• Question 4

Prove that (n + 10)2 - (n + 2)2 is a multiple of 16 for all positive values of n.

Write your full proof on a separate piece of paper then record your final concluding statement below.

Do not use a space between any of the numbers or signs in your final expression, or you may be marked incorrectly.

16(n+6)
16(n + 6)
16 (n + 6)
16(n+ 6)
16(n +6)
EDDIE SAYS
Remember in proof questions you need to expand and simplify the expressions to be able to rearrange them into a form that can be factorised. If we expand the brackets, we reach: n² + 20n + 100 - n² - 4n - 4 If we simplify this expression, the n² terms cancel each other out, to reach: 20n + 100 - 4n - 4 = 16n + 96 Finally, we need to factorise this expression to show that 16 is always a multiple: 16(n + 6)
• Question 5

Prove that (4n + 1)2 - (4n - 1)2 is a multiple of 8 for all positive values of n.

Write your full proof on a separate piece of paper then record your final concluding statement below.

Do not use a space between any of the numbers or signs in your final expression, or you may be marked incorrectly.

8(2n)
8( 2n )
8 (2n)
8 ( 2n )
EDDIE SAYS
Remember in proof questions you need to expand and simplify the expressions to be able to rearrange them into a form that can be factorised. Be careful with the signs here. Write the expansion of the second bracket in bracket first: 16n² + 8n + 1 - (16n² - 8n + 1) Then check which signs need to be changed: 16n² + 8n + 1 - 16n² + 8n - 1 If we simplify this expression, we reach: 16n We then need to factorise to show that 8 is a multiple of this expression: 8(2n) How did you do with that one?
• Question 6

Prove that (2n + 1)(n + 3) + (2n + 1)(n - 2) is an odd number.

Write your full proof on a separate piece of paper then record your final concluding statement by choosing from the options below.

2(2n² + n) + 1
EDDIE SAYS
This question was a little harder. After expanding and simplifying brackets, you need to spot how to factorise the expression to show the desired result. The question asks you to show that the number is odd. Odd numbers are always one more (or one less) than an even number. Let's follow our steps: 1) Expand the brackets: 2n² + 6n + n + 3 + 2n² - 4n + n - 2 2) Simplify: 4n² + 2n + 1 3) Factorise to show that the outcome is odd: 2(2n² + n) + 1 One more than an even number (a multiple of 2 or 2n) is odd. Does that make sense?
• Question 7

Show algebraically that the product of two even numbers is always a multiple of 4.

Write your full proof on a separate piece of paper then record your final concluding statement by choosing from the options below.

4(n2)
EDDIE SAYS
Read the question carefully. The key part is 'product of two even numbers'. Take a symbol for two even numbers (either the same or different, it doesn't matter) and multiply them together: e.g. 2n(2n) You want to show that you get a multiple of 4, so factorise with 4 outside of the bracket: 4n2 = 4(n2) Done!
• Question 8

Prove algebraically that the sum of two consecutive whole numbers is always odd.

Write your full proof on a separate piece of paper then record your final concluding statement by choosing from the options below.

2n + 1
EDDIE SAYS
Here you're not asked to add ('sum') odds or evens specifically, so use n for any whole number. Add n and the next number, n+1: n + (n + 1) When we simplify, we get: 2n + 1 We know that 2n represents an even number, so (n + 1) is one more than an even number, so it must be odd.
• Question 9

Prove algebraically that the difference between the squares of two consecutive odd numbers is always even.

Write your full proof on a separate piece of paper then record your final concluding statement below.

Do not use a space between any of the numbers or signs in your final expression, or you may be marked incorrectly.

2(4n+4)
2(4n + 4)
2 (4n+4)
2 (4n + 4)
EDDIE SAYS
Read the question carefully. 'Difference' means subtracting. You need to use two consecutive odd numbers: 2n+1, 2n+3. So our starting statement should be: (2n + 3)² - (2n + 1)² Be careful of the signs here: 4n² + 6n + 6n + 9 - (4n² + 2n + 2n + 1) = 4n² + 6n + 6n + 9 - 4n² - 2n - 2n - 1 Let's simplify this now: 8n + 8 Now we need to factorise to prove that the outcome is always even. Remember that we use '2n' to denote an even number, so we need to factorise by 2 in our proof: 2(4n + 4)
• Question 10

True or false?

12n always represents an even number.

True
EDDIE SAYS
This statement is true. We usually use 2n to denote an even number. 12n can be rewritten as a multiple of 2, like this: 2(6n) This means that it is always an even number. Great work on this activity as this is not an easy concept! Why not think of some of your own mathematical statements which you can prove or disprove now for some extra practise? 