Mathematical Induction

This is the section of my presentation!!

 1. Definition:

mathematical induction is a method for proving that statements involving natural numbers are true for all natural numbers.  

2. Theorem:

suppose that 他the following two conditions are satisfied with regard to a statement bout natural numbers:

CONDITIONＩ:The statement is true for  natural number 1

Condition II: if the statement is true for some natural number k, it is also true for k+1

Thus the statement is true for all numbers

3. sums of powers of integers

Examples:

1.

For any positive integer n, 1 + 2 + ... + n = n(n+1)/2.

           n=1  1 = 1(2)/2 which is clearly true.

          Here we must prove the following assertion: "If there is a k such that P(k) is true, then (for this same k) P(k+1) is true."

        assume there is a k such that 1 + 2 + ... + k = k (k+1)/2.  

         We must prove, for this same k, the formula 1 + 2 + ... + k + (k+1) = (k+1)(k+2)/2.

        1 + 2 + ... + k + (k+1) = k(k+1)/2 + (k+1) = (k(k+1) + 2 (k+1))/2 = (k+1)(k+2)/2.

2.

Write the first five terms of the sequence

a0=1 an=a(n-1)+2

a0=1

a1=3

a2=5

a3=7