Unit 1: Number and Algebra

Topic 1.6: Proof Questions

Practice 20 exam-style questions for IB Math AA SL Topic 1.6. Review the question stems below, then unlock the full Question Bank to access markschemes, model answers, and AI grading.

1Prove3 marks

2026

Prove that the sum of any two even numbers is even.

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2Choose1 mark

In a proof,

n

stands for any integer. Which expression always represents an odd integer?

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3Sketch1 mark

2022

It is given that

2 \times \frac{n ( n + 1 )}{2} \equiv n (n + 1)

.

For

n = 3

, sketch a dot diagram that shows this result.

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4Choose1 mark

Which factorisation shows most directly that

7 n + 21

is a multiple of 7 for every integer

n

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5Choose1 mark

Two consecutive integers are added together. If the smaller is

n

, which expression always equals their total?

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6Choose1 mark

Expand and simplify

(x + 3) (x + 7)

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7Choose1 mark

To prove the identity

(x + 4) (x - 4) \equiv x^{2} - 16

, what should the working do?

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8Show3 marks

2026

Show that

(n + 2)^{2} - n^{2} = 4 (n + 1)

for all integers n.

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9Prove2 marks

2026

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

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10Show2 marks

2026

Show that 5n + 15 is a multiple of 5 for every integer n, and state the integer it is 5 times.

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11Prove2 marks

2026

Prove the identity

(x + 1) (x + 5) \equiv x^{2} + 6 x + 5

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12Find1 mark

2026

Find the value of

5 3^{2} - 5 1^{2}

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13Write down1 mark

2026

Write down expressions for the other two.

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14Write down1 mark

2026

Write down the value of the expression when

x = 2.5

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15Choose1 mark

Three consecutive integers are written as

n

n + 1

and

n + 2

. Which expression gives their sum in factorised form?

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16Sketch1 mark

2026

Sketch the graph of

y = (x + 3)^{2} - (x - 3)^{2}

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17State1 mark

2022

It is given that

\frac{n ( 3 n - 1 )}{2} \equiv \frac{n ( n + 1 )}{2} + n (n - 1)

for

n \in Z^{+}

, where

\frac{n ( 3 n - 1 )}{2}

is the

n

th pentagonal number and

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

is the

n

th triangular number.

State in words what this identity shows.

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18State1 mark

2022

It is given that

1 + 3 + 5 + \dots + (2 n - 1) \equiv n^{2}

for

n \in Z^{+}

.

State in words what this identity shows about the odd numbers.

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19Prove2 marks

2026

Prove the identity (x + 1)(x + 5) ≡ x² + 6x + 5.

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20Prove2 marks

2026

Prove the difference-of-squares identity a² − b² ≡ (a + b)(a − b).

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