Mathematical Induction презентация

numbers n.
Suppose that
1. S1 is true.
2. If Sk is true, then Sk +1 is also true.

Then Sn is true for all positive integer numbers n.

sequence {xn} converges, then

Question 1. Let x1 = 1 and

Solution. First, we find fixed points of the relationship

That is, we find solutions of the equation

Next, we calculate a few terms of the sequence.

for any positive integer n.

Our first statement S1:

The general statement Sn:

The first statement S1 is correct.
Let us assume that a statement Sk : is also correct. Then

free-response question?
Then we can pass to the limit

Since we know that the sequence xn converges

Since is a continuous function for we obtain

4:

Question 3a:

Question 3b:

Integer part of

Question 5:

You have to show that

equivalence

we obtain

result for this limit.

To obtain a correct result we have to use

Incorrect!

equivalent functions for each term at x = 0

Therefore,

S1 = (0,0).
The first 3 turning points, keeping the snail confined inside a unit square, are given by S2 = (½, 1), S3 = (1, ½), and S4 = (½, 0).
After that the snail always heads towards the midpoint of the next path segment that it sees, without crossing its own path.
That is, the coordinates (xn, yn) of the turning points Sn are given by

and

for n = 5,6,7,…

= 2,
and find a similar relationship for the y-coordinates.
b) Find the x and y coordinates of the limiting point of the snail path.
c) Repeat parts a) and b) for a snail confined inside an isosceles triangle.
The initial location of the snail is S1 = (0,0).
The first 2 turning points are S2 = (¾, ½) and S3 = (½, 0).

equation has only two real solutions, but it definitely has a solution

Divide the characteristic polynomial by

Denote

then

the limit in the relationship

we obtain

Hence

= 20182 – 1.