IB · Mathematics: Analysis & Approaches SL & HL · Number & algebra

Proof: IB Mathematics: Analysis & Approaches questions with answers

Free Proof questions for IB Mathematics: Analysis & Approaches (SL & HL), part of Number & algebra. Each question has the answer and the marking points an examiner would look for.

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Proof practice questions

Question 1Structured · 4 marks

Prove that the product of three consecutive integers, added to the middle integer, is equal to the cube of the middle integer.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
proof
let the integers be n − 1, n, n + 1 M1
(n − 1)n(n + 1) + n = n(n2 − 1) + n A1
= n3 − n + n A1
= n3, the cube of the middle integer R1
4
Question 2Structured · 5 marks
(a)

Show that 12r − 1 − 12r + 1 = 2(2r − 1)(2r + 1).

[2]
(b)

Hence find the exact value of Σr=191(2r − 1)(2r + 1).

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)shown
(2r + 1) − (2r − 1)(2r − 1)(2r + 1) M1
= 2(2r − 1)(2r + 1) AG A1
2
(b)919
12Σr=1n(12r − 1 − 12r + 1) M1
telescoping: 12(1 − 119) (A1)
919 A1
3
Question 3Structured · 4 marks

Prove that the sum of any four consecutive integers is never divisible by 4.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
proof
n + (n + 1) + (n + 2) + (n + 3) M1
= 4n + 6 A1
= 4(n + 1) + 2 A1
which leaves remainder 2 on division by 4, so is never divisible by 4 R1
4
Question 4Structured · 5 marks
(a)

Show that 12r − 1 − 12r + 1 = 2(2r − 1)(2r + 1).

[2]
(b)

Hence find the exact value of Σr=1491(2r − 1)(2r + 1).

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)shown
(2r + 1) − (2r − 1)(2r − 1)(2r + 1) M1
= 2(2r − 1)(2r + 1) AG A1
2
(b)4999
12Σr=1n(12r − 1 − 12r + 1) M1
telescoping: 12(1 − 199) (A1)
4999 A1
3
Question 5Structured · 4 marks

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

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
proof
let the integers be n and n + 1 M1
(n + 1)2 − n2 = n2 + 2n + 1 − n2 A1
= 2n + 1 A1
= n + (n + 1), the sum of the integers R1
4
Question 6Structured · 5 marks
(a)

Show that 12r − 1 − 12r + 1 = 2(2r − 1)(2r + 1).

[2]
(b)

Hence find the exact value of Σr=1191(2r − 1)(2r + 1).

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)shown
(2r + 1) − (2r − 1)(2r − 1)(2r + 1) M1
= 2(2r − 1)(2r + 1) AG A1
2
(b)1939
12Σr=1n(12r − 1 − 12r + 1) M1
telescoping: 12(1 − 139) (A1)
1939 A1
3

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Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in