Mathematical Induction
Mathematical Induction - How it Works - Video
Definitions
Definitions:
The principle of mathematical induction needs to have two statements so that it works.
Statement 1 - Pn is true.
Statement 2 - Whenever k is a positive integer such that Pk is true, then Pk+1 is also true.
Steps in Applying:
The principle of mathematical induction needs to have two statements so that it works.
Show - Statement 1 - Pn is true.
Assume - Statement 2 - Whenever k is a positive integer such that Pk is true, then Prove Pk+1 is also true.
Example 1 - Explore
Example 1 Explore:
If are not sure what to first, then try plugging in some numbers to see if you can see a pattern. After substituting 1, 2, and 3 we found that the statement 1 + 2 + 3 + ... + n = [n * (n + 1)] / 2.
The 1st term
=> 3 *(-1/4)1-1
=> 3 *(-1/4)0
=> 3 * 1
=> 3
The 2nd term
=> 3 *(-1/4)2-1
=> 3 *(-1/4)1
=> 3 * (-1/4)
=> (-3/4)
The 3rd term
=> 3 *(-1/4)3-1
=> 3 *(-1/4)2
=> 3 * (1/16)
=> 3/16
The 10th term
=> 3 *(-1/4)10-1
=> 3 *(-1/4)9
=> 3 * (-1/262,144)
=> -3/262,144
Example 1 Step 1 & Step 2
Example 2:
Luckily for us, when we explored, we already did the first step and prove that P1 is true. Now we substitute k into our statement and assume that is true. Finally we want to prove k + 1 is true so we down our goal.
Example 1 Step 2:
Example 1 Step 2:
Here we have all the steps to get our assumption to our goal.
1st => We put parentheses around the first times before k + 1
2nd => We substituted the induction hypothesis
3rd => We found a common denominator.
4th => We group the fractions together.
5th => We factored the (k + 1) term since it is on both sides of the addition sign.
6th => We changed (k + 2) to match our goal.
Example 1 Final Part:
Example 1 Final Part:
We have shown that P1 is true and we proving Pk+1 is true.
Therefore we can say that Pk+1 is true by mathematical induction.