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Methods of Proof · Guided Practice

Methods of Proof

20 questions with answers and video solutions. Try each one before revealing the answer.

1
Find a counterexample to show that this statement is false: ∀n∈R,n2=n.\displaystyle \forall n \in \mathbb R, \sqrt{n^{2}} = n.
2
Find a counterexample to show that the following conjecture is false:
Let Pn\displaystyle P_n represent the product of the first n\displaystyle n prime numbers. Then Pn+1\displaystyle P_{n}+1 is prime ∀n∈N.\displaystyle \forall n \in \mathbb N.
3
Prove that if a\displaystyle a is a multiple of 2\displaystyle 2 and b\displaystyle b is a multiple of 3\displaystyle 3 then ab\displaystyle ab is a multiple of 6.\displaystyle 6.
4
Prove that the sum of the squares of two odd numbers is even.
5
Prove that any multiple of 3\displaystyle 3 can be expressed as the sum of three consecutive integers.
6
Use proof by contradiction to demonstrate that 2\displaystyle \sqrt{2} is irrational.
7
Use proof by contradiction to show that there is an infinite number of prime numbers.
8
Use the contrapositive to prove that if n2\displaystyle n^2 is a multiple of 3\displaystyle 3 then n\displaystyle n is a multiple of 3.\displaystyle 3.
9
Prove by contrapositive that if pq\displaystyle pq is irrational then at least one of p\displaystyle p or q\displaystyle q is irrational.
10
Prove by induction that ∀n∈N,6n+4\displaystyle \forall n \in \mathbb N, 6^{n}+4 is divisible by 10.\displaystyle 10.
11
The Fibonacci sequence is defined by the recurrence relation: F1=F2=1\displaystyle F_1=F_2=1 and Fn+2=Fn+1+Fn (n≥1).\displaystyle F_{n+2}=F_{n+1}+F_{n}\ (n \ge 1).
Prove by induction that, ∀n∈N,F1+F2+⋯+Fn=Fn+2−1.\displaystyle \forall n \in \mathbb N, F_1+F_2+\dots +F_n=F_{n+2}-1.
12
2016 Q5
4 Marks

Prove by induction that ∑r=1nr(3r−1)=n2(n+1)\displaystyle \sum_{r=1}^{n}r(3r - 1) = n^2(n + 1) , ∀n∈N.\displaystyle \forall n \in \mathbb{N}.

13
2019 Q14
5 Marks

Prove by induction that

∑r=1nr!r=(n+1)!−1\displaystyle \sum_{r=1}^{n}r!r = (n+1)! - 1

for all positive integers n.\displaystyle n.

14
2023 P1 Q8
(1, 2)3 Marks

(a)Consider the statement:
For all integers a\displaystyle a and b\displaystyle b, if a<b\displaystyle a < b then a2<b2.\displaystyle a^2 < b^2.
Find a counterexample to show that the statement is false.

(b)Let n\displaystyle n be an odd integer.
Prove directly that n2−1\displaystyle n^2 - 1 is divisible by 4.

15
2022 P1 Q6
(1, 3)4 Marks

Consider the statement: For all odd numbers n\displaystyle n, n2+4\displaystyle n^2 + 4 is prime.

(a)Find a counterexample to show that the statement is false.

(b)Prove directly that the difference between the cubes of any two consecutive integers is not divisible by 3.

16
2023 P2 Q12
5 Marks

Prove by induction that, for all positive integers n\displaystyle n,

∑r=1n2r−1r=2n(n−1)+1.\displaystyle \sum_{r=1}^{n} 2^{r-1}r = 2^n(n-1)+1.

17
2024 P2 Q11
3 Marks

Consider statements A and B below.
For each statement: if true, provide a proof; if false, provide a counterexample.

A: The sum of the squares of any two consecutive integers is always prime.
B: The sum of the squares of any two consecutive integers is always odd.

18
2025 P2 Q15
5 Marks

Prove by induction that, for all positive integers n\displaystyle n,

∑r=1n1(2r+1)(2r−1)=n2n+1.\displaystyle \sum_{r=1}^{n} \frac{1}{(2r + 1)(2r - 1)} = \frac{n}{2n + 1}.

19
2026 P2 Q12
5 Marks

Prove by induction that 7n+2\displaystyle 7^{n}+2 is divisible by 3 for all n∈N.\displaystyle n\in\mathbb{N}.

20
2026 P2 Q15
(1, 3)4 Marks

Let r\displaystyle r be a positive real number and consider the following statement:

If r\displaystyle r is irrational then r\displaystyle \sqrt{r} is irrational.

(a)Write down the contrapositive of the statement.

(b)Hence prove that the statement is true.

Practice questions courtesy of Maths.scot. Full written solutions are on his site — the links above go straight to them.

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