MATHEMATICAL INDUCTIONMIDWESTERN STATE UNIVERSITY

WHY THE DROPLET TEMPLATE???

Because Mathematical induction makes most students cry!

Will try to make it painless.

PRINCIPLE OF MATHEMATICAL INDUCTION

Let S(n) be a statement involving an integer n.

Suppose that, for some fixed integer n0,

a) S(n0) is true (i.e. S(n) is true if n = n0 )

b) When integer k >= n0 & S(k) is true, then S(k+1) is true

The S(n) is true for all integers n >= n0

EXAMPLE

S(n): 1 + 2 + 3 +… + n = (n (n + 1)) / 2

• n=4 (4*5)/2 = 10

• n=5 (5*6)/2 = 15

• n=10 (10 * 11)/2 = 55

• n=100 (100*101)/2 = 5050

INDUCTION SAYS….

S(n): 1 + 2 + 3 +… + n = (n (n + 1)) / 2

a) Show that S(n) is true for some (small) integer k

a) That means it is true at least sometimes!

b) Since S(n) is true for some integer k, show it is also true for k+1

a) That means it is true for all integers >= k

LETS PROVE IT USING INDUCTION

S(n): 1 + 2 + 3 +… + 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

CONTINUINGS(n): 1 + 2 + 3 +… + 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): 1 + 2 + 3 +… + 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

TO REPEAT MYSELF

1.Show that n = 1 works in the formula2.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

ANOTHER PROBLEMProve 1 + 3 + 5 + . . . + (2n-1) =n2

Clearly 1 works so assume that k works so lets prove that k+1 will work

1 + 3 + 5 +…+ (2k-1) + 2(k+1)-1= k2 + 2k + 1

= (k+1)2

QED

IN CLASS EXERCISES

Prove: For n > 1, 1+4+9+…+n2= n(n+1)(2n+1)/6

Prove: For n > 1,   2 + 22 + 23 + 24 + ... + 2n = 2n+1 – 2

Prove: For n > 5, 4n < 2n

HOMEWORK

• Page 84 – Problems 12 & 13