Algebraic Proof (Edexcel IGCSE Maths A: Higher)

Exam Questions

2 hours39 questions
1
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3 marks

Prove algebraically that

         left parenthesis 2 n space plus space 1 right parenthesis squared space minus space left parenthesis 2 n space plus space 1 right parenthesis is an even number

for all positive integer values of n.

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2
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3 marks

Show that open parentheses n plus 3 close parentheses squared space minus space open parentheses n minus 3 close parentheses squared  is an even number for all positive integer values of n.

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3
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4 marks

n is an integer greater than 1

Prove algebraically that  n squared space minus space 2 space minus space open parentheses space n minus 2 close parentheses squared is always an even number.

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4
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3 marks

Prove that the difference between two consecutive square numbers is always an odd number.
Show clear algebraic working.

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5
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3 marks

N is a multiple of 5

table row cell A space end cell equals cell space N space plus space 1 end cell row cell B space end cell equals cell space N space – space 1 end cell end table

Prove, using algebra, that A squared – space B squared is always a multiple of 20

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6
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2 marks

E space equals space n squared plus space n space plus space 5

Ali thinks that the value of E will be a prime number for any whole number value of n.

Is Ali correct?
You must give a reason for your answer.

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7
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4 marks

p is a positive number.

n is a negative number.

For each statement, tick the correct box.

 

Always true

Sometimes true

Never true

p plus n is positive

square

square

square

p minus n is positive

square

square

square

p squared plus n squared is positive

square

square

square

p cubed divided by n cubed is positive

square

square

square

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8
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4 marks

x is an integer.

Prove that  35 space plus space left parenthesis 3 x space plus space 1 right parenthesis squared space – space 2 x left parenthesis 4 x space – space 3 right parenthesis  is a square number.

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9
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1 mark

Which of these is a correct identity?

Circle your answer.

x space plus space 4 x space identical to space 5 x

6 x space identical to space 18

2 x space plus space 1 space identical to space 7

7 x space plus space 9 space identical to space x

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10
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3 marks

space k space equals space n squared space plus space 9 n space plus space 1

Mo says,    “k will be a prime number for all integer values of n from 1 to 9”

Show that Mo is wrong.
You must show that your value of k is not prime.

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11
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1 mark

Tick whether the following statement is true or false.

Give a reason for your answer.

When n space is a positive integer, the value of 2 n is always a factor of the value of 20 n.

True    begin mathsize 26px style square end style      False    begin mathsize 26px style square end style

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12
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4 marks

Prove that the mean of any four consecutive even integers is an integer.

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13a
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3 marks

Prove that the sum of four consecutive whole numbers is always even.

13b
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2 marks

Give an example to show that the sum of four consecutive integers is not always divisible by 4.

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1
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3 marks

Prove that

open parentheses 2 n space plus space 3 close parentheses squared space minus space open parentheses 2 n space minus space 3 close parentheses squared  is a multiple of 8 

for all positive integer values of n

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2
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4 marks

Prove that the square of an odd number is always 1 more than a multiple of 4

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3
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4 marks

Prove that , for all positive values of n,

fraction numerator open parentheses n space plus 2 space close parentheses squared space minus space open parentheses n space plus 1 close parentheses squared over denominator 2 n squared space plus space 3 n end fraction space equals space 1 over n

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4
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4 marks

Prove algebraically that the difference between the squares of any two consecutive integers is equal to the sum of these two integers.

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5
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4 marks

Prove algebraically that the product of any two odd numbers is always an odd number.

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6a
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1 mark

Show that x left parenthesis x space minus space 1 right parenthesis space open parentheses x space plus space 1 close parentheses equals space x cubed space minus space x

6b
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3 marks

Prove that the difference between a whole number and the cube of this number is always a multiple of 6

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7
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4 marks

Prove algebraically that the difference between the squares of any two consecutive odd numbers is always a multiple of 8

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8
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4 marks

n is the middle integer of three consecutive positive integers.

The three integers are multiplied to give a product.

n is then added to the product.

Prove that the result is a cube number.

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9
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4 marks

Expressions for consecutive triangular numbers are

fraction numerator n open parentheses n plus 1 close parentheses over denominator 2 end fraction and fraction numerator open parentheses n plus 1 close parentheses open parentheses n plus 2 close parentheses over denominator 2 end fraction

Prove that the sum of two consecutive triangular numbers is always a square number.

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10
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3 marks

n is a positive integer.

Prove algebraically that  2 n squared open parentheses 3 over n plus space n close parentheses space plus space 6 n space open parentheses n squared space minus 1 close parentheses is a cube number.

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11
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2 marks

a squared space – space b squared space identical to space left parenthesis a space plus space b right parenthesis left parenthesis a space – space b right parenthesis

    a and b are positive whole numbers with a space greater than space b

    a squared space – space b squared is a prime number.

Why are a and b consecutive numbers?

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12
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3 marks

c is a positive integer.

Prove that  fraction numerator 6 c cubed plus 30 c over denominator 3 c squared plus 15 end fraction  is an even number.

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13
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3 marks

The lengths of the sides of a right-angled triangle are all integers. Prove that if the lengths of the two shortest sides are even, then the length of the third side must also be even.

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14a
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2 marks

Express as a single fraction.

fraction numerator m plus 1 over denominator n plus 1 end fraction minus m over n

Simplify your answer.

14b
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2 marks

Using your answer to part (a), prove that if m and n are positive integers and m space less than space n, then

fraction numerator m plus 1 over denominator n plus 1 end fraction minus m over n greater than 0

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1
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3 marks

(i) Factorise 2 t squared plus 5 t plus 2.

[2]

(ii) t is a positive whole number.

The expression 2 t squared plus 5 t plus 2 can never have a value that is a prime number.
Explain why.

[1]

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2
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2 marks

n is an integer.

Prove algebraically that the sum of  1 half space n left parenthesis n space plus space 1 right parenthesis and 1 half space left parenthesis n space plus space 1 right parenthesis left parenthesis n space plus space 2 right parenthesis is always a square number.

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3
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6 marks

Here are the first five terms of an arithmetic sequence.

7      13      19      25      31 

Prove that the difference between the squares of any two terms of the sequence is always a multiple of 24.

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4
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2 marks

Given that n can be any integer such that n space greater than space 1, prove that n squared space minus space n is never an odd number.

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5
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3 marks

The product of two consecutive positive integers is added to the larger of the two integers.

Prove that the result is always a square number.

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6
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3 marks

Prove that when the sum of the squares of any two consecutive odd numbers is divided by 8, the remainder is always 2
Show clear algebraic working.

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7
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3 marks

Using algebra, prove that, given any 3 consecutive whole numbers, the sum of the square of the smallest number and the square of the largest number is always 2 more than twice the square of the middle number.

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8
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3 marks

Using algebra, prove that, given any 3 consecutive even numbers, the difference between the square of the largest number and the square of the smallest number is always 8 times the middle number.

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9a
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2 marks

Here are the first four terms of a sequence of fractions.

1 over 1 space space space space space space 2 over 3 space space space space space space 3 over 5 space space space space space space space 4 over 7

The numerators of the fractions form the sequence of whole numbers 1 space 2 space 3 space 4 space...
The denominators of the fractions form the sequence of odd numbers 1 space 3 space 5 space 7 space...

Write down an expression, in terms of n, for the n th term of this sequence of fractions.

9b
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3 marks

Using algebra, prove that when the square of any odd number is divided by 4 the remainder is 1

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10
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4 marks

The table gives information about the first six terms of a sequence of numbers.

  Term number

1

2

3

4

5

6

  Term of sequence

fraction numerator 1 space cross times space 2 over denominator 2 end fraction space

fraction numerator 2 space cross times space 3 over denominator 2 end fraction space

fraction numerator 3 space cross times space 4 over denominator 2 end fraction space

fraction numerator 4 space cross times space 5 over denominator 2 end fraction

fraction numerator 5 space cross times space 6 over denominator 2 end fraction

fraction numerator 6 space cross times space 7 over denominator 2 end fraction

Prove algebraically that the sum of any two consecutive terms of this sequence is always a square number.

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11
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3 marks

Prove that x squared space plus space x space plus space 1 is always positive.

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12a
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2 marks

The diagram shows a cross placed on a number grid.

A grid of numbers 1 to 60 in sequential order. Numbers 25, 34, 35, and 36 are shaded to highlight them distinctively.

L is the product of the left and right numbers of the cross.
T is the product of the top and bottom numbers of the cross.
M is the middle number of the cross.

Show that when M space equals space 35 comma space L space – space T space equals space 99.

12b
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5 marks

Prove that, for any position of the cross on the number grid above, L space – space T space equals space 99.

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