Mathematical Induction

1. Mathematical Induction

2. Consider the following statement • Any n squares can be cut into pieces and then pasted into one single square. • Let us denote this statement by S(n). • Of course n  2. • First of all, let us consider S(2):

5. Conclusion • Hence S(2) is true, that is, any 2 squares can be cut into pieces and then pasted into one single square. • But how about other statements? • So, let’s go on to S(3).

6. By S(2) By S(2)  S(3) is true.

7. By S(2) By S(3)  S(4) is true.

8. By S(2) By S(4)  S(5) is true.

9. S(3) S(4) S(5) Conclusion • Hence, we can see that S(n) is true for all integers n  2 because: 1. S(2) is true, 2. If S(k) is assumed to be true, then S(k+1) is true. S(2)

10. Principle of Mathematical Induction • Let S(n) be a statement about a positive integer n  1. Then S(n) is always true if: 1. S(1) is true. 2. If S(k) is assumed to be true, then S(k + 1) is true.

11. Example To prove: For all n  1, 1 + 2 + 3 + … + n = Proof: When n = 1, L.H.S. = 1 R.H.S. = L.H.S. = R.H.S. when n = 1.

12. Next, assume that LHS = RHS when n = k, that is, 1 + 2 + 3 + … + k = Then, when n = k + 1, LHS = 1 + 2 + 3 + … + k + (k + 1)

13. Conclusion Hence, by principle of M.I., LHS = RHS for all n  1.

14. Worksheet • Show that for any natural number n, • 12 + 22 + 32 + … + n2 =