Download 9.4 Notes - Jessamine County Schools

9.4 MATHEMATICAL INDUCTION (1)
- Mathematical Induction is a method of proving that statements involving natural numbers are
true for all natural numbers
- Recall that the natural numbers are the numbers 1, 2, 3, 4,5 ,…
The Principle of Mathematical Induction
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Suppose that the following two conditions are satisfied with regards to a statement about
natural numbers:
o Condition 1: The statement is true for the natural number 1
o Condition 2: If the statement is true for some natural number k, it is also true for the
next natural number k+1
Then the statement is true for all natural numbers.
We can use the idea that a collection of dominoes, lined up one after the other, represents the
collection of natural numbers. Suppose we are told tow facts:
1) The first domino is pushed over
2) If one domino falls over, say the kth domino, then so will the next one, the (k+1)st domino.
It is safe to conclude that all the dominoes fall over if the first falls (Condition 1), then the second one
does also (Condition 2); and if the second falls over, then so does the third (by Condition 2); and so on.
Ex) Show that the following statement holds true for all natural numbers n:
1) Show holds for n = 1
Statement hold for n= 1 (Condition 1)
2) We now assume that the statement hold for some natural number k
Substitute k for n
Called the ASSUMPTIVE STATEMENT
3) We now need to show the statement holds for some natural number k+1
Sub (k+1) for all the k’s
Simplify using distribution
Using the principle counting, we can count up to a number k by using the idea of
Counting every other number
The bolded part of the equation
represents the ASSUMPTIVE
STATEMENT from above. We
substitute
for this part.
Simplify statement into a quadratic
Factor the expression
Equal sides
4) Thus, by Mathematical Induction, the statement holds true for all natural numbers.
** This statement must be written in order to finalize the proof as being completed **
Ex) Show that the following statement holds true for all natural numbers n:
1) Show holds for n = 1
Statement hold for n= 1 (Condition 1)
2) We now assume that the statement hold for some natural number k
Substitute k for n
Called the ASSUMPTIVE STATEMENT
3) We now need to show the statement holds for some natural number k+1
Sub (k+1) for all the k’s
Simplify
Using the principle counting, we can count up to a number k by using the idea of
Counting every other number
The bolded part of the equation
represents the ASSUMPTIVE
STATEMENT from above. We
Substitute
for this part.
Combine fractions
Expand terms
Combine like terms
Equal sides
4) Thus, by Mathematical Induction, the statement holds true for all natural numbers.
** This statement must be written in order to finalize the proof as being completed **