User:Elif Tokay/sandbox
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Proof by Induction[edit]
Proof by Induction is one of the most known proofs that are used in mathematics to show if an equation is working for the defined set of numbers or not. However, unlike most of the other proofing methods, inductional proof can only be used in prooving formulas that works calculations on aritmetic strings, and usually integers. The reason why inductional proof doesnt work on irregular series is that it is based on the similarity between the connection of numbers. Usually the methot is applied by 3 steps.
Steps[edit]
1) Choosing a referance point and a range of numbers the prove the formula in.
2) Showing that the expressions works on the referance number.
3) Assuming the formula works on the number "n", then prooving it world also work on the number "n+(the determined aritmatic range of numbers)"
The logic of the steps depends on the idea that every time it is assumed that "n" is a working number for the formula, it becomes a referance point to prove the validity of "n+(aritmatic range)" and makes it the next referance point to prove the next number on the string. This proving methot is mostly used in positive integers because of the aritmetic feature and the easily proovable referance point of it.
Example[edit]
n(n+1)/2 is a well known formula for calculating the addition of integers from 1 to n. Inductional proof of this formula starts with prooving it ensures for the integer 1.
For n=1, n(n+1)/2 = 1(1+1)/2 = 1.2/2 = 1 Since the addition of all integers from 1 to 1 is 1, the formula ensures the first condition.
For the second condition, it will be assumed that x(x+1)/2 is the addition of all integers from 1 to x and then it is to be proved that the addition of integers from 1 to x+1 is n(n+1)/2 where n=x+1
If n=x+1, n(n+1)/2 = (x+1)(x+1+1)/2 = (x+1)(x+2)/2
To finish up, we will check if the addition of integerrs from 1 to x+1 is equal to the addition of integers from 1 to x plus x+1.
(x+1)(x+2)/2 = x(x+1)/2 + (x+1)
If we cancel (x+1) from all multiplications it will leave us with;
(x+2)/2 = x/2 + 1
And if multiplied by 2;
x+2 = x+2
Which proves the formula is correct.