MATHEMATICAL INDUCTION

1. Use the method of mathematical induction to prove the following important results:

(a)

(b)

(c)

(d)

2. Use the principle of mathematical induction to prove that:

(a)

(b)

3. Prove, by mathematical induction, that:

(a)

(b)

4. Use the method of mathematical induction to prove that:

(a)

(b)

5. Determine the formula for the sum to n terms, and use the method of mathematical induction to prove these formulae:

(a) 1 + 4 + 7 + 10 + . . . (b) 1 + 4 + 42 + 43 + . . .

6. Prove that:

(a) is divisible by 4.

(b) is divisible by 80.

(c) is divisible by *x *- 1.

7. Show that:

(a)

(b)

8. Prove the following identities by finite (methamatical) induction for integers n __>__ 1

(a) If and then .

(b)

[hints: , say]

9. Show by mathematical induction that:

(a)

(b) where p>-1

10. Use the method of mathematical induction to prove that:

(a) is diivisible by 5. [hints: ]

(b) is divisible by 10, if n is odd [hints: in step 2, assuming the result is true for n = k, where k is odd, then we have to show that it is true for the next odd integers (n = k+2) ].

*Prove the following by using the method of mathematical induction:*

11.

12.

13.

14. The sum of *n* terms of an arithmetic sequence whose first term is *a *and common difference *d* is

15. for *x* > 0.

16. is divisible by 3.

17.

18. *n*(*n* + 1) is an even number.

19. 2 + 4 + 6 + . . . . . + 2*n* = *n*(*n* + 1)

20. for *n *__>__ 1.

21. for *n* > 4.

22. *n*(*n* + 1) + (*n* + 2) is divisible by 3.

23.

24.

25. is divisible by (*x* - 1)

26. for all *x* __>__ 1

27. is divisible by 5.

28. The sum of the cubes of three consecutive integers is divisible by 3.

29.

30.

31.

32. for all *x* __>__ 1

33. for *x* __>__ 2.