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Saturday, 7 January 2017

Mathematical Induction

Mathematical Induction

Mathematical induction is a form of mathematical proof.
Just because a rule, pattern, or formula seems to
work for several values of n, you cannot simply
decide that it is valid for all values of n without
going through a legitimate proof.

The  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 integer k,
then Pn  must be true for all integers n.

Ex.  Use mathematical induction to prove the following
        formula.
Sn = 1 + 3 + 5 + 7 + . . . + (2n-1) = n2
First, we must show that the formula works for n = 1.

1.For n = 1
  S1 = 1 = 12
The second part of mathematical induction has two
steps.  The first step is to assume that the formula is
valid for some integer k.  The second step is to use this
assumption to prove that the  formula is valid for the
next integer,  k + 1.

2.Assume Sk = 1 + 3 + 5 + 7 + . . . + (2k-1) = k2
is true, show that Sk+1 = (k + 1)2   is true.

Sk+1 = 1 + 3 + 5 + 7 + . . . + (2k – 1) + [2(k + 1) – 1]  
= [1 + 3 + 5 + 7 + . . . +(2k – 1)] + (2k + 2 – 1)
= Sk + (2k + 1)
= k2 + 2k + 1
=(k + 1)2
To read complete topic ...................click here



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