Principle of Mathematical Induction

Let Pn be a statement involving the positive integer n.
   If
   1.  P1 is true, and
   2.  the truth of Pk implies the truth of Pk-1,  for every positive k,
  then
   Pn must be true for all positive integers n.

Examples:

1.  Prove:

     Proof:
        i.  Prove the statement is true for n = 1.
           1 =  3(12) - 1   = 3(1) - 1  =  2   =  1
                         2                  2         2

      ii.  Assume the statement is true for n = k.

     iii.  Prove the statement is true for n = k + 1.

           In other words, prove

           Start with the assumption in ii.

           Add the next term in the sequence to both sides.

          If the original statement is true, then


         (The left side is a result of applying the statement to the (k + 1)st term and the right side is the right side of the
          above equation.)

         Simplify the right side.



         Now simplify the left side.

         The two expressions are equal so the statement is true.
 
 

Finite Differences

The first differences of a sequence are found by subtracting consecutive terms.  If these are all the same, the sequence has a linear model.
The second differences are found by subtracting consecutive first differences.  If these are all the same, the sequence has a quadratic model.

Find the quadratic model for   3,5,8,12,17,23,…

Sequence:             3     5     8    12     17     23     ...
1st differences:          2     3    4     5        6          ...
2nd differences:            1    1     1       1              ...

We know we have a quadratic model because the 2nd differences are all the same.
To find the model,  use n as x, and use an as y.
   y = ax2 + bx + c    3 = a(1)2 + b(1) + c    5 = a(2)2 + b(2) + c     8 = a(3)2 + b(3) + c
                                3 = a + b + c               5 = 4a + 2b + c             8 = 9a + 3b + c

   Use elimination to solve:
     Subtract:
     5 = 4a + 2b + c                    8 = 9a + 3b + c
     3 = a +     b + c                    5 = 4a + 2b + c
     2 = 3a + b                            3 = 5a + 5b

    Multiply the first answer by -5 and add it to the second answer:
     -10 = -15a - 5b
        3 =    5a + 5b
      -7 = -10a
      7/10 = a

     Substituting the value for a into one of the two sentences in a and b:
           3 = 5(7/10) + 5b
           3 = 3.5 + 5b
         -.5 = 5b
         -.1 = b
     -1/10 = b

    Substituting the values of a and b back into one of the first equations:
           3 = a + b + c
           3 = .7 - .1 + c
           3 = .6 + c
        2.4 = c
    24/10 = c
      12/5 = c

    So    y = .7x2 -.1x + 2.4

Problems

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