Algebraic argument and proof

Here are two expressions, A and B.

Show that A and B are equivalent.

f(x) = ax + b and g(x) = where a and b are positive integers.

Prove that fg(x) – af^–1(x) is always a multiple of 3

x is a positive even number. y = (x – 3)(x – 5)(x + 6)

Without expanding brackets, explain why there is only one value of x for which y is negative.

c and d are positive numbers. c is even. d is odd.

Tick a box for each expression.

9k + 7 and 2k² + 3 are consecutive integers. 9 k + 7 is the smaller integer.

Work out the value of the next consecutive integer.

x is a square number.

3(x – 1) 3x – 3 is an identity. Tick one box.

4 3(x – 1) 3x – 3 is an identity. Tick one box.

Here is the term-to-term rule for a sequence.

The first three terms of the sequence are a + 1 2 a + 5 4 a + 13

Show that the sum of the first four terms is a multiple of 3

Here is the term-to-term rule for a sequence.

The first three terms of the sequence are a + 1 2 a + 5 4 a + 13

Show that the sum of the first four terms is a multiple of 3

By completing the square, prove that x² + 6x + 13 is always positive.

Here is a number machine.

Show that when the input increases by 2 the output increases by 2 a.

² f(x) = kx where k is a constant. Kai says that is equal to f(3) because = 3 Is he correct? Show working to support your answer.

Two integers have a difference of 6 The integers are multiplied together. 9 is then added. Prove algebraically that the result is always a square number.

n is an odd number. Why is n (n + 1) always an even number?

Circle the correct statement.

12x³ + 7x² + 3x – 10 ≡ 2(ax³ + x² + 2x – 5) + x(bx + c) Work out the values of a, b and c.

x is an integer. Prove that 35 + (3x + 1)² – 2x(4x – 3) is a square number.

Which of these is a correct identity? Circle your answer.

Here is an identity. a(3x – 10) 21x + 2b Work out the values of a and b.

Here is an identity. a(3x – 10) 21x + 2b Work out the values of a and b.

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.

Here is an identity.

Use the identity to work out the value of 193² – 7² You must show your working.

Factorise 100a² – 81b²

Expressions for consecutive triangular numbers are

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

Two consecutive whole numbers are n and n + 1

Simplify n – (n + 1)

Multiply out n (n + 1)

The two numbers are added. Show that the answer must be an odd number.

Work out the values of a and b in the identity

n is a positive integer. Prove algebraically that 2n² n 6n n² 1 is a cube number.

3 a + b = 7 and 6 x + 8 y = 40 Show that 9 a + 3 b has a greater value than 3 x + 4 y

Megan says, “If you add any three multiples of 10 the total must be a multiple of 10 and a multiple of 3” Is she correct? You must show your working.

a² – b² ≡ (a + b)(a – b) a and b are positive whole numbers with a > b a² – b² is a prime number. Why are a and b consecutive numbers?

Prove that x² + x + 1 is always positive.