sec. 2 Proof

Question 1

What are the 4 rules of proof?

Answer 1

Question 2

Prove that the sum of any three odd numbers is odd.

Answer 2

Question 3

Prove that the (n+3)² - (n-2)² ≡ 5(2n+1).

Answer 3

Question 4

R says ‘’ the diff. between any two consecutive numbers square numbers is always a prime number ‘’.

Prove R is wrong.

Answer 4

Question 5

Prove that the difference between 10¹⁸ and 6²¹ is a multiple of two.

Answer 5

Question 6

The range of a set of +ive numbers is 5. Each number in the set is doubled.

Show that the range of the new set of numbers also doubles.

Answer 6

Question 7

Ellie says, ‘‘If x > y, then ‘‘x > y’’.

Is she correct? Explain your answer.

Answer 7

Question 8

Prove that the sum of the ext. angles of a triangle is

360°

Answer 8

Question 9

Prove that (n+3)² - (3n+5) ≡ (n-3)(n+2)+2

Answer 9

Question 10

Prove that (n-3)² - (n-5) ≡ (n-3)(n-4)+2

Answer 10

Question 11

Prove that 25 - (x-8)²/4 ≡ (2+x)(18-x)/4

Answer 11

Question 12

Prove that if 3(ax + 7) - 2( x+b) ≡ 4x + 29, then a=2 and b=-4

Answer 12

Question 13

Prove that (2n+1) - (2n-1) - 10 is not a multiple of 8 for +ive values of n.

Answer 13

Question 14

Prove that the sum of any three consecutive numbers is divisible by 3

Answer 14

Question 15

Prove that an even number multiplied by another even number will always result in an even number.

Answer 15

Question 16

Prove that the sum of any three consecutive even numbers is always a multiple of six

Answer 16

Question 17

Prove these statements:

a) The sum of any two consecutive odd numbers must always be a multiple of 4
b) The sum of the squares of any two consectice odd numbers cannot be a multiple of 4.

Answer 17

Question 18

Maisie says ‘‘all pime numbers are odd.’’

Prove M is wrong.

Answer 18

2 is a prime number and it is even.

Question 19

Prove that these statements are wrong:

a) If the sum of two integers is even, one of the integers must be even.
b) If n is an integer and n² is divisible by 4, then n is also divisible by 4 .

Answer 19

Question 20

C says ‘‘if a²=b²’’, then ‘‘a=b’’.

Prove that she is wrong.

Answer 20

Question 21

Prove that 5²⁰ - 5¹⁹ is even without using a calc.

Answer 21

Question 22

Without using a calculator, prove the sum of 21⁸ and

15⁴ is a multiple of 3.

Answer 22

Question 23

Without using a calculator, prove that 3⁸ - 1 is not a prime number.

Answer 23

Question 24

The nth term of a sequence is given by 1/2n - 5/2n+3.

Prove that the sum of any two consecutive numbers in a sequence is is a square no.

Answer 24

Question 25

If a= 5⁹⁹ x 2⁹⁸ and b= 5⁷² x 4³⁸, show that ab has 172

digits when written as an ordinary number

Answer 25

Question 26

WB

Answer 26

Question 27

The mean of a set of 6 no.s is 18.

If 1 is subtracted from each number in the set, show that the mean of the new set also also decreases by 1.

Answer 27

Question 28

The nth term of a sequence is given by n - 2n + 2.

Prove that the sum of any two consecutive no.s in the sequence is an odd number.

Answer 28

Question 29

Fay claims that x² + 3 > 2x + 1 for all the values of x.

Prove that Fay is correct.

Answer 29

Question 30

EPW

Prove that (3n+2)

Question 31

Jake says ‘‘if a

Is he correct? Explain your answer.

Question 32

Prove that the diff. between the squares of two consecutive even no.s is always a multiple of 4.

Question 33

Show that the number 2⁶⁴ -1 is not prime