Topic 1 - Proofs

Question 1

Prove that x^2 - 4x + 9 is always positive

Answer 1

x^2 - 4x + 9 = (x - 2)^2 + 5
(x - 2)^2 is always positive as x^2 is always positive for any x. The +5 will retain the positivity, and therefore x^2 - 4x + 9 is always positive.

Question 2

Prove that every perfect cube number is a multiple of 9, one less than a multiple of 9 or one more than a multiple of 9.

Answer 2

Every integer n is a multiple of 3 (n=3m), one less than a multiple of 3 (n=3m-1) or one more than a multiple of 3 (n=3m+1).
Consider n=3m, then
n^3=(3m)^3=27m^3=9 x 3m^3 which is a multiple of 9. Now consider n=3m-1, then
n^3=(3m-1)^3=27m^3-27m^2+9m-1=9(3m^3-3m^2+m)-1 which is one less than a multiple of 9. Finally consider n=3m+1, then
n^3=(3m+1)^3=27m^3+27m^2+9m+1=9(3m^3+3m^2+m)+1 which is one more than a multiple of 9 and concluding the proof by exhaustion.

Question 3

Prove that (n+1)^3 ≥ 3^n for 
n ∈ ℕ, n ≥ 4.

Answer 3

We must show that (n+1)^3 is greater than 3n for all n that is a natural number less than or equal to 4. The natural numbers less than or equal to 4 include 1, 2, 3, and 4. Proving the above by exhaustion will involve showing that it is true for 1, 2, 3 and 4:

• For n=1, (n+1)^3 = (1+1)^3 = 2^3 = 8 and 3^n = 3^1 = 3. Since, 8 ≥ 3, the above is true when n=1.
• For n=2, (n+1)^3(2+1)^3 = 3^3 = 27 and 3^n = 3^2=9. Since, 27 ≥ 9, the above is true when n=2.
• For n=3, (n+1)^3 = (3+1)^3 = 4^3 = 64 and 3^n = 3^3 = 27. Since, 64 ≥ 27, the above is true when n=3.
• For n=4, (n+1)^3 = (4+1)^3 = 5^3 = 125 and 3^n = 3^4 = 81. Since, 125 ≥ 81, the above is true when n=4 and thus concluding the proof by exhaustion.

Question 4

What does the symbol ‘ℕ’ mean?

Answer 4

Natural numbers - positive integers

Question 5

What does the symbol ‘ℤ’ mean?

Answer 5

Integers

Question 6

What does the symbol ‘ℚ’ mean?

Answer 6

Rational numbers

Question 7

What does the symbol ‘∈’ mean?

Answer 7

………….. is an element of …………..

Question 8

What does the symbol ‘ℝ’ mean?

Answer 8

Real numbers