1. Mathematical Induction
Midwestern State University

2. Why the droplet Template???
Because Mathematical induction makes most students cry! Will try to make it painless.

3. Principle of Mathematical induction
Let S(n) be a statement involving an integer n. Suppose that, for some fixed integer n0,
S(n0) is true (i.e. S(n) is true if n = n0 )
When integer k >= n0 & S(k) is true, then S(k+1) is true
The S(n) is true for all integers n >= n0

4. Example S(n): 1 + 2 + 3 +… + n = (n (n + 1)) / 2 n=4  (4*5)/2 = 10

5. Induction Says…. S(n): 1 + 2 + 3 +… + n = (n (n + 1)) / 2
Show that S(n) is true for some (small) integer k
That means it is true at least sometimes!
Since S(n) is true for some integer k, show it is also true for k+1
That means it is true for all integers >= k

6. Lets Prove it using induction
S(n): … + n = (n (n + 1)) / 2 Lets first show S(n) is true for some small integer. In this case lets use 1. If we plug in 1 we get 1 = (1(1+1))/2 and hence we see that 1 =1 This means that 1 works in the formula for n

7. Continuing
S(n): … + n = (n (n + 1)) / 2 Now assume that n works. So giving that assumption lets see if S(n+1) works. We can do this by plugging in n+1 for n in the formula. SO! We start with the LHS and generate the RHS using pure algebra! Capice!? S(n+1): … + n + n+1= (n (n+1))/2+ n+1 = (n (n+1))/2+ 2( n+1)/2 = ((n (n+1))+ 2( n+1))/2 = ((n + 2)( n+1))/2 = ((n + 1)+ 1)( n+1))/2 = S (n+1) QED

8. To REPEAT MYSELF
Show that n = 1 works in the formula
Assume that n works in the formula and show with this
assumption that n+1 works
This will prove that ALL of the n values will work

9. another Problem
Prove (2n-1) =n2 Clearly 1 works so assume that k works so lets prove that k+1 will work …+ (2k-1) + 2(k+1)-1= k2 + 2k + 1 = (k+1)2 QED

10. IN Class Exercises Prove: For n > 1, 1+4+9+…+n2= n(n+1)(2n+1)/6

11. Homework
Page 84 – Problems 12 & 13