MATH 122
MATHEMATICAL INDUCTION PROBLEMS

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

  1.     13 + 23 + 33 + . . . + n3 =

  2.     =
  3.     =
  4.     2 + 6 + 10 + ... + 4n - 2 = 2n2
  5.     21 + 22 + 23 + ... + 2n = 2n + 1 - 2
  6.     =

  7.     =

By induction show that:
  1.     3n - 1 is divisible by 2.
  2.     5n - 1 is divisible by 4.
  3.     7n - 1 is divisible by 6.
  4.     82n - 1 is divisible by 63.
  5.     62n - 1 is divisible by 35.
  6.     92n - 1 is divisible by 80.
  7.     n2 – 3n +4 is even.
  8.     2n3 – 3n2 + n is divisible by 6.
  9.     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?