Unit
Sequences and Series
University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
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University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
An important and fundamental tool used when doing proofs is mathematical induction. We can use mathematical induction to prove properties in math, or formulas. For example, we can prove that a formula works to compute the value of a series. Mathematical induction involves using a base case and an inductive step to prove that a property works for a general term.
So at some point in your Math career you will have or will have or have seen these formulas, okay? And what they are are basically sums of different numbers. You are looking at the just the sum of bunch of integers and we know that the sum is going to be n n+1 over 2. We also have a formula for the sum of squares and the sum of cubes. And on the surface these formulas look pretty complicated, okay. But actually how we can prove that they are true is by using what we call mathematical induction, okay? And what mathematical induction is is basically we prove it works for n=1. So we show that these equations all hold for n=1. We then assume that the equation holds for n=k some arbitrary k and then sorry, using that fact we show that it's true for n is equal to k+1. Okay? And basically how that works is if it works for 1, we can assume it works for any number. k could be any number. So k could be 1 as well. So if it works for 1, then this would show that it works for 2. If it works for 2 then we already know it works for the next one. It works for 3, it works for 4 [IB] 5 so on and so forth, okay?
So mathematical induction is basically a type of approach to proving a statement. You show it works for your first term, you assume it works for some arbitrary variable, typically k is used and then using the fact that it works you're assuming it works for k, you prove that the equation still holds for and is equal to k+1.