Browse Algebraic proof

41 questions at your level

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

Expressing generality — the language of proof

18 questions

LessonNot started

n

is a whole number. Write down an expression for the next whole number after

n

●●●●●Level 11 markStart

n

is a whole number. Explain why

2n + 1

is always odd.

●●●●●Level 11 markStart

n

is an integer.

(a) Write down an expression, in terms of

n

, for any odd number.

(b) Write down expressions for the next two odd numbers after

2n + 1

●●●●●Level 22 marksStart

Prove that the sum of any three consecutive integers is a multiple of 3.

●●●●●Level 32 marksStart

n

is an integer.

(a) Write down an expression, in terms of

n

, for any even number.

(b) Explain why

2n + 1

is an odd number for every integer

n

●●●●●Level 32 marksStart

Prove that the sum of any odd number and any even number is odd.

●●●●●Level 33 marksStart

Prove that the sum of any two consecutive odd numbers is a multiple of 4.

●●●●●Level 43 marksStart

Tomas says, "The sum of any two prime numbers is always even."

Show that Tomas is wrong.

●●●●●Level 42 marksStart

Prove that

n^2 + n

is even for every integer

n

●●●●●Level 43 marksStart

Aisha says, "The difference between the squares of any two consecutive integers is always even."

Show that Aisha is wrong.

●●●●●Level 42 marksStart

Prove that the sum of any four consecutive odd numbers is a multiple of 8.

●●●●●Level 44 marksStart

Prove that the product of any two consecutive even numbers is a multiple of

8

. (4 marks)

●●●●●Level 44 marksStart

Prove that

(n + 3)^2 - (n - 1)^2

is a multiple of

8

for all integer values of

n

. (3 marks)

●●●●●Level 43 marksStart

(a) Factorise

m^2 - n^2

(1 mark)
(b) Show that

5^6 - 1

is the product of two consecutive even numbers.

(2 marks)

●●●●●Level 43 marksStart

There are only lemon cupcakes and chocolate cupcakes on tray A.

number of lemon cupcakes : number of chocolate cupcakes

= 2 : 7

Tray B holds only vanilla cupcakes and coffee cupcakes.

Tray B holds a third as many cupcakes as tray A.

There are

x

lemon cupcakes on tray A.

Use algebra to show that, altogether, trays A and B hold

6x

cupcakes.

(3 marks)

●●●●●Level 43 marksStart

Prove that the product of any two odd numbers is odd.

●●●●●Level 54 marksStart

3a : 2c = 4 : 5

b : 4c = 3 : 2

Show that

a + b : b + c = 14 : 15

(3 marks)

●●●●●Level 53 marksStart

There are only green pens and black pens in a box.

The number of green pens is

3

times the number of black pens.

Zara takes at random one pen from the box, records its colour and puts it back in the box.

She does this

n

times (

n \geqslant 2

).

Write down an expression in

n

for the probability that she takes at least one green pen and at least one black pen.

(2 marks)

●●●●●Level 52 marksStart

Proving divisibility results

13 questions

LessonNot started

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

●●●●●Level 33 marksStart

Prove that the sum of any three consecutive odd numbers is a multiple of 3.

●●●●●Level 43 marksStart

Prove that

(3n + 1)(n + 2) - n(3n + 7) = 2

for all values of

n

●●●●●Level 43 marksStart

Prove that

(2n + 3)(n + 1) - n(2n + 5) = 3

for all values of

n

●●●●●Level 43 marksStart

Prove that the sum of any five consecutive even numbers is a multiple of 10.

●●●●●Level 43 marksStart

Prove that the product of any two consecutive even numbers is a multiple of

8

. (4 marks)

●●●●●Level 44 marksStart

Prove that

(n + 3)^2 - (n - 1)^2

is a multiple of

8

for all integer values of

n

. (3 marks)

●●●●●Level 43 marksStart

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

8

. [4]

●●●●●Level 44 marksStart

Prove that the difference between the squares of any two consecutive even numbers is a multiple of 4.

●●●●●Level 53 marksStart

Prove that

n^3 - n

is a multiple of 6 for every integer

n

●●●●●Level 54 marksStart

Prove that the sum of any four consecutive integers is never a multiple of 4.

●●●●●Level 53 marksStart

Prove that

n^2 + 3n

is even for every integer

n

●●●●●Level 53 marksStart

Prove that

n^3 + 3n^2 + 2n

is a multiple of 6 for every integer

n

●●●●●Level 54 marksStart

Proofs with squares

14 questions

LessonNot started

Prove that

(n + 4)^2 - n^2

is a multiple of 8 for every integer

n

●●●●●Level 33 marksStart

Prove that

(n + 3)^2 - (n - 3)^2

is a multiple of 12 for every integer

n

●●●●●Level 43 marksStart

Prove that the square of any odd number is odd.

●●●●●Level 43 marksStart

Prove that

(3n + 1)^2 - 1

is a multiple of 3 for every integer

n

●●●●●Level 43 marksStart

Prove that

x^2 + 4x + 7

is positive for all values of

x

●●●●●Level 43 marksStart

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

8

. [4]

●●●●●Level 44 marksStart

The

n

th triangular number is

\dfrac{n(n + 1)}{2}

Prove that 8 times any triangular number, plus 1, is always a square number.

(3 marks)

●●●●●Level 43 marksStart

(a) Factorise

m^2 - n^2

(1 mark)
(b) Show that

5^6 - 1

is the product of two consecutive even numbers.

(2 marks)

●●●●●Level 43 marksStart

Prove that

x^2 - 6x + 10

is positive for all values of

x

●●●●●Level 53 marksStart

Prove that the sum of the squares of any two consecutive integers is odd.

●●●●●Level 53 marksStart

Priya says, "

n^2 + n + 11

is a prime number for every positive integer

n

."

Show that Priya is wrong.

●●●●●Level 52 marksStart

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

●●●●●Level 54 marksStart

Ellie says, "

n^2 - n + 5

is a prime number for every positive integer

n

."

Show that Ellie is wrong.

●●●●●Level 52 marksStart

Prove that the sum of the squares of any two consecutive odd numbers is always

2

more than a multiple of

8

. (4 marks)

●●●●●Level 54 marksStart