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Use Formal Algebraic Proof

In this worksheet, students will learn how to use algebraic proof.

'Use Formal Algebraic Proof' worksheet

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:  

Worksheet Overview

QUESTION 1 of 10

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

 

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

Expand the brackets and simplify:

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

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

= 4n + 12

Factorise to show that 4 is a factor and so 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:

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

Sum means adding. We are going to add three consecutive even numbers (2n, 2n+2, 2n+4):

2n + 2n + 2 + 2n + 4

Simplify:

6n + 6

Factorise to show 6 is a factor:

6(n + 1)

Match the expressions to the correct descriptions.

Column A

Column B

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

Pick the term which means multiplying.

Sum

Difference

Product

Quotient

Pick the correct labels to go with the symbols/expressions.

 odd numbermultiple of 3consecutive numberseven 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.

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

n is an integer.

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

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

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

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

True or False?

12n represents an even number.

True

False

  • Question 1

Match the expressions to the correct descriptions.

CORRECT ANSWER

Column A

Column B

an odd number
2n + 1
a multiple of 7
7n
consecutive even numbers
2n, 2n+2, 2n+4
any number
n
consecutive odd numbers
2n+1, 2n+3, 2n+5
an even number
2n
EDDIE SAYS
n is any number. If you want to have an even number, it must be 2n, whereas an odd number will be 2n+1 (1 more than an even number). Have a look at the table in the introduction again if you need to.
  • Question 2

Pick the term which means multiplying.

CORRECT ANSWER
Product
EDDIE SAYS
Product means we need to multiply. Sum is adding, difference is subtracting and quotient is dividing.
  • Question 3

Pick the correct labels to go with the symbols/expressions.

CORRECT ANSWER
 odd numbermultiple of 3consecutive numberseven number
3n
n n+1 n+2
2n+1
2n
EDDIE SAYS
3n is a multiple of 3. n, n+1, n+2 are consecutive numbers. 2n+1 is an odd, 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.

CORRECT ANSWER
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. Expand and simplify (n + 10)² - (n + 2)² and then factorise with 16 outside of the bracket. This will show that the expression is a multiple of 16.
  • Question 5

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

CORRECT ANSWER
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 This will ensure that you end up with the correct expression.
  • Question 6

n is an integer.

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

CORRECT ANSWER
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. This is why I factorised into the form 2(2n² + n) + 1: multiples of 2 are all even, but we have an extra + 1 at the end making the whole expression odd.
  • Question 7

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

CORRECT ANSWER
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. You want to show that you get a multiple of 4, so factorise with 4 outside of the bracket. Done!
  • Question 8

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

CORRECT ANSWER
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. When you simplify, you get 2n+1 which is one more than an even number, so it must be odd.
  • Question 9

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

CORRECT ANSWER
EDDIE SAYS
Read the question carefully. 'Difference' means subtracting. You need two consecutive odd numbers: 2n+1, 2n+3. The first one is smaller, so take its square away from the square of the second one. Be careful with expanding brackets. A grid could help you with that. Remember to factorise with a 2 outside of the brackets at the end to show that it is an even number.
  • Question 10

True or False?

12n represents an even number.

CORRECT ANSWER
True
EDDIE SAYS
This is true. Even though we usually use 2n to show an even number, 12n can be rewritten as a multiple of 2: 2(6n), so it is an even number.
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