Write an explanation of the idea behind mathmatical induction.?

Question

Please!

Answer 1

Mathematical induction is NOT the same as ordinary inductive reasoning, where we simply consider a number of cases, observe a pattern, and induce that this pattern is true for all such cases. In math we have to be more rigorous. The pattern for mathematical induction is:
1. Show that a proposition is true for n = 0 or n = 1
2. Show that if the propostion is true for n,
then it's true for n+1
Mathematical induction is used in a lot of proofs in number theory. Here's an exerpt from Wikipedia:
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"This method works by first proving the statement is true for a starting value, and then proving that the process used to go from one value to the next is valid. If these are both proven, then any value can be obtained by performing the process repeatedly. It may be helpful to think of the domino effect; if you have a long row of dominoes standing on end, and you can be sure that:
The first domino will fall
Whenever a domino falls, its next neighbor will also fall,
then you can conclude that all of the dominoes will fall, and this fact is inevitable."
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You can read the rest of the article here..
http://en.wikipedia.org/wiki/Mathematical_induction
The above site also has a few examples of mathematical induction. For more examples, see
http://www.cut-the-knot.org/induction.shtml

Answer 2

The idea is that one way to structure a proof is the "divide and conquer" strategy, dividing a more difficult question into several less difficult questions.

We must ASSUME that:
For any x (in lower level mathematics x would be a number)
and for the proposition in question (call it P ) which we want to prove. We ASSUME that:
A. IF we can show that IF P is true for x+1 when it is true for x
and

B. If we can show P is true for a specific x'

then it must true for all x (greater than x').

As far as I know, but I'm no mathmatician nor philosopher, you can't prove that the ASSUMPTION is TRUE, you can however, easily show it works well, so it is a good (even if not true) assumption.
But you can see where we've taken a proof of "P is true for all greater or equal to x' " and split it into two easier (hopefully) proofs.

Answer 3

Mathematical induction requires you to show two things:
1) That the statement is true for n = 1
2) If the statement holds for n = k, then it also holds for n = k + 1
If this is so, then the statement holds for all n.

This is why: You showed it held for n = 1. You also showed that if it holds for a number, then it holds for that number plus 1. So it also holds for 1 + 1 = 2. Thus, it holds for 2 + 1 = 3, and so on.

Answer 4

Supopose you have to prove T(n), n an integer , T a statement

for some n = N you have proofed that T(N)

if you can proove T(N+1) you have proved for all n>=N by indiuction