I am just starting to learn about Mathematical proofs and so far I have learned about Mathematical Induction. I would like to know in its core, what is the main conceptual difference between proving something using Induction and Deduction.
Example to illustrate my question:
Let's say I wanna prove that the following statement is true:
1 + 3 + 5 + ... + (2n-1) = n²
Using Induction:
We prove for the base, thus, when n = 1
Left = 1 | Right = 1² = 1
Therefore it works for my base case
Now, by assumption: n = k where k is a natural number
Then: 1 + 2 + 3 +...+ (2k-1) = k²
If it is true, it should work for n = k + 1, thus: 1 + 2 + 3 +...+ (2k-1) + (2k + 1) = k² + (2k + 1)
k² + (2k + 1) can be factroized as (k + 1)²
So we've just prove that the sentence works for n = k + 1, therefore, it works for any natural number.
Now let's prove by deduction:
We can easily recognize the series {a} = 1 + 2 + 3 + ... + (2n-1) as the sum of odd numbers and it is an Arithmetic Progression in its core, being the (2n-1) the General Term. We know that the general form for summing up an Arithmetic Progression is: S = (n/2) × [2a + (n−1)d]
In this case a is equals to 1 and d is equals to 2 Replacing in the formula, we get: (n/2) × (2n) which is n²
Then we found out by Algebra that the statement is true
So, What is the difference between using the two methods?
Thanks in advance
Well it can be said that deduction is using given data and some acquired knowledge or rules to prove that a certain property is true, so it is arriving to a result by logical reasoning.
Induction ofcourse also uses logical reasoning, but it is used when you want to prove that a certain property is true for any natural number $n$. Also even if you want to prove that a property is true for any $n$, the method of induction differs than that of deduction as in your example. Induction proves that a property is true for $n=1$ first(or $n=0$ or ... according to the property you're proving), assumes its true up to order $n$, and then proves that it is true for $n+1$, by doing that you will have proved that your property is true for any $n$. Deduction, on the other hand, directly proves the property for any $n$ using some rules.