Mathematical Induction ( with video)
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January 07, 2017 |
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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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