Can someone explain Mathematical Induction to me please?

Question

I don't get how to do things like: Use mathematical induction to prove the statement, and they give you something like n on top of the sigma, i=1 on the bottom, and on the right is (2i+1) = n(n+2)

Answer 1

You have this sum:

When i = 1, then 2i+1 = 3

When i = 2, then 2i+1 = 5,

etc.

The sum sign symbolises this, the i values. There are from 1 to n, you are supposed to know what n is.

Example:

sum (1 to 4) (2i+1) = 3 + 5+ 7+ 9

In this case, n = 4

The result is this: n(n+2). Since n = 4, then you have 4(4+2) = 4*6 = 24

Lets add now and see if this functions:

3+5+7+9

3+5 = 8

8 + 7 = 15

And 15 + 9 = 24

This functions for n = 4

Since you dont want to calculate forall possible n, then you prove it by induction. Induction seems a domino game.

Imagine this domino pieces:

llllllllllllll....llll.......lllll.....

If the first one falls, then the second falls. If the second falls, then the third falls, ...if the nth piece falls, then the n+1 falls, etc.

Then all of them fall.

If you can prove this:

1) that the first one falls
2) if the nth falls, then the (n+1)th falls,

you have proved that all of them fall.

Hope this helps

Ana

Answer 2

The Principle of Mathematical Induction (PMI) says that if you have a statement P(n) about a number n, and you can prove:
(a) P(1) is true
(b) For all positive integers n, if P(n) is true then P(n+1) is also true
then P(n) is true for every positive integer n.

You can see this makes sense: by applying (b) repeatedly we get P(2) is true, P(3) is true, P(4) is true... and so on.

(An equivalent form of (b) is to prove that for all positive integers n, if P(k) is true for all 1<= k <=n, then P(n+1) is true.)

So, in the example:
P(n) is the statement Σ(i=1 to n) (2i+1) = n(n+2)
P(1) is the statement Σ(i=1 to 1) (2i+1) = 1(1+2)
<=> 2(1) + 1 = 3 which is true.
Now suppose P(n) is true. We want to show that P(n+1) is also true. The trick here is to break down the sum of (n+1) terms into the sum of the first n terms plus the last term.
P(n+1) says
Σ(i=1 to n+1) (2i+1) = (n+1)((n+1)+2)
<=> Σ(i=1 to n) (2i+1) + 2(n+1)+1 = (n+1)(n+3)
<=> n(n+2) + 2n + 3 = n^2 + 4n + 3 [using the assumption that P(n) is true, so Σ(i=1 to n) (2i+1) = n(n+2).]
<=> n^2 + 4n + 3 = n^2 + 4n + 3 which is true.
So P(1) is true and for any positive integer n, if P(n) is true then P(n+1) is true. Hence by PMI, P(n) is true for every positive integer n.

Answer 3

Mathematical induction Mathematical induction is a ability of mathematical evidence in many circumstances used to ensure that a given fact is real of all organic numbers. it truly is carried out by ability of proving that the 1st fact interior the countless sequence of statements is real, and then proving that if anybody fact interior the countless sequence of statements is real, then so is the subsequent one. the technique could be prolonged to coach statements approximately greater favourite properly-based systems, jointly with timber; this generalization, common as structural induction, is used in mathematical good judgment and computing gadget technological know-how. The validity of mathematical induction is logically equivalent to the properly-ordering concept. Mathematical induction should not be misconstrued as considered one of those inductive reasoning, it is seen non-rigorous in arithmetic. (See concern of induction.) in fact, mathematical induction is taken into consideration one of those deductive reasoning and is totally rigorous.