Mathematical Proofs

WMA12/01 IAL (Edexcel)/P2/Jan 2021/Q5 (Proof by Deduction; Counter Example)

• Use algebra to prove that for all​​ x≥ 0​​ 

3x + 1≥2 3x 

(3)

• Show that the following statement is not true.​​ 

“The sum of three consecutive prime numbers is always a multiple of 5”​​ 

(1)

SOLUTION

i-​​ 

First, we are going to use jotting method to find a good starting point, but remember, jotting is not the part of the proof.​​ 

Jotting

3x + 1≥2 3x 

3x+12≥23x2

9x2+6x+1≥43x

9x2+6x+1≥12x 

9x2-6x+1≥0

3x-12≥0

Proof​​ 

We always start a proof from a known or settled fact like in this case, we know that the expression squared is always​​ ≥​​ 0.​​ 

Consider,​​ For add real x values 

3x-12≥0

(3x)2-23x1+12≥0

9x2-6x+1≥0

9x2-6x+1+12x≥12x

9x2+6x+1≥12x

3x+12≥12x

Taking square root on both sides.

3x+12=12x

Since​​ x≥ 0,​​ 12x​​ will be​​ +12x.

3x+1=4×3x

3x+1=23x

Hence, proved.

ii- Lets consider the three consecutive numbers.  ​​​​ 

For 2, 3, and 5,

2+3+5=10 (10 is a multiple of 5) 

For 3, 5, and 7,

3+5+7=15 (15 is a multiple of 5)

For 5, 7, and 11,

5+7+11 =23 (23 is not a multiple of 5)

Hence, the sum of three consecutive prime numbers (5+7+11) isn’t a multiple of 5.