MATH 122

MATHEMATICAL INDUCTION PROBLEMS

Use induction to prove that each of the following formulas is true for each positive integer n.

1

^{3}+ 2^{3}+ 3^{3}+ . . . + n^{3}=- =
- =
- 2 + 6 + 10 + ... + 4n - 2 = 2n
^{2} - 2
^{1}+ 2^{2}+ 2^{3}+ ... + 2^{n}= 2^{n + 1}- 2 - =
=

By induction show that:

- 3
^{n}- 1 is divisible by 2. - 5
^{n}- 1 is divisible by 4. - 7
^{n}- 1 is divisible by 6. - 8
^{2n}- 1 is divisible by 63. - 6
^{2n}- 1 is divisible by 35. - 9
^{2n}- 1 is divisible by 80. - n
^{2}– 3n +4 is even. - 2n
^{3}– 3n^{2}+ n is divisible by 6. - a. Show: If 2 + 4 + 6 + ... + 2n = n(n
+ 1) + 2 is true for n = j, then it is true for n = j + 1.

b. Is the formula true for all n?