Is mathematical induction a deductive or inductive argument?

[jeremy22511]

Homework Statement

Is mathematical induction a deductive or inductive argument?
Would appreciate the help. Thanks.

Jeremy

Homework Equations

The Attempt at a Solution

Its name suggests that the process is inductive, yet I know all of mathematics depends on deductive reasoning...

 

[tjackson3]

Short answer: Your hunch was dead on. It's deductive reasoning (inductive is not accepted as a valid type of reasoning in most disciplines).

Long answer: The first step in mathematical induction is inductive. I'll use as an example the formula for summing all consecutive integers from 1 to n:

1 + 2 + ... + n = \frac{n(n+1)}{2}

The first step in proving this is to prove that it's true for n = 1. That is:

1 = \frac{2}{2} = 1

To stop there would be to use inductive reasoning - i.e., since it's true for n = 1, it must be true for all n. This is obviously not necessarily correct, and that's where the deductive part comes in. The purpose of a deductive argument is to prove that, given a hypothesis, its conclusion must be valid and follow directly from the hypothesis. That is, now that we know that the above is true for n = 1, we assume that it's true for some n (that's the hypothesis), and show that it must then be true for n + 1. Now that it's in general form like that, you've completed the deduction, and shown that it's true for all n in the domain of the problem (in this case, natural numbers).