1 formal proof

Question 1

How do you prove the sum of a geometric sequence?

Answer 1

• Step 1 set out sn with sn=a+ar+ar^2+ar^3…ar^n-1
  -Step 2 set out rsn with rsn=ar+ar^2+ar^3… ar^n
  -Step 3 sn - rsn
  Step 4 then factor out and put it in form

Question 2

How do you prove that x/y + y/x > 2 for all positive real values.

Answer 2

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Question 3

How do you prove that x/y + y/x > 2 for all positive real values.

Answer 3

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Question 4

How do you prove that there are infinitely many prime numbers?

Answer 4

-Assume there is a finite number of prime numbers
-in the form p1, p2, p3,…pn.
-Now consider number K=p1,p2,p3,…pn+1
This means when p1,p2,p3….pn+1 is divided by any of the prime numbers with remainder 1.

-This implies that either K is prime or K has a prime factor that is not listed.

Question 5

How do you prove that root 2 is irrational?

Answer 5

-Assume root 2 is rational in its simplified form where a and b have no factors.
-then show that a is even
a=2n
so b^2=2n^2
so be is even.

as it is even/even there are common factors which contradicts initial assumption.

Question 6

how could you prove that the statement that a linear line is sometimes, never or always greater than a mod.

Answer 6

-Sketch the graphs and show that there is a region where they are above each other.

Question 7

What does R mean?

What does Q mean?

What does Z mean?

What does N mean?

Answer 7

• All real numbers eg 2pi
• Rational numbers 1/10
• Whole numbers
• Natural numbers 1,2,3

Question 8

How could you prove that for all natural numbers that n^2+2 is divisible by 4.

Answer 8

-model real numbers as either odd or even then sub them in and show they are not divisible.

Question 9

How could you show that a modulus is greater than or equaled to a linear line?

Answer 9

-Use graphical approach and explain.

Question 10

How do you know when you get vertical asymptotes on a curve?

Answer 10

-Vertical asymptotes when denominator = 0

Question 11

How to deal with questions that want you to find ff(x) and involve subbing in fraction and fraction?

Answer 11

-Deal with the numerator and denominator separately, simplify as much as possible, then skip flip multiply at the end.

Question 12

How do you prove that n^2-n is always even

Answer 12

-Factorise and then suggest the different possible outcomes.

Question 13

How do you prove that any number is irrational?

Answer 13

-First statement must be a and be have no common factors