In this lesson, our instructor Vincent Selhorst-Jones teaches Geometric Sequences and Series. In this lesson youll learn about the form for the nth term and the formula for geometric series. Youll also learn about infinite geometric series, including diverges and converges. The lesson ends with five fully worked out examples.
One of the most important ideas in mathematics is being able to prove something with total certainty. We can show not just that something works out, but that it will work every time, forever-it is mathematical fact. We show this with proof.
Mathematical induction is a form of proof that allows us to prove that a pattern holds true forever.
A good metaphor for the idea of induction is a never-ending line of dominoes. If we have two guarantees,
The first domino will fall over;
If a domino falls over, it will cause the next domino to fall over as well;
then we know that all the dominoes in our infinitely long line of dominoes will fall over.
The principle of mathematical induction is basically the same idea as the above metaphor, except we replace "domino falling over" with "statement being true". Let P1, P2, P3, P4, ..., Pn,
... be some sequence of statements where n is a positive integer. If
P1 is true, and
If Pk is true for a positive integer k, then Pk+1 must also be true,
then the statement Pn is true for all positive integers n.
Above, we call the first step the base case because it establishes the basis we are working from.
Above, the second step is called the inductive step since that is where the actual induction occurs. In the inductive step, the assumption "if Pk is true" is called the inductive hypothesis, because we must use this hypothesis
to show Pk+1 will also be true.
It helps a lot to somehow include part of the statement for Pk when we are writing out Pk+1. This is because we will almost inevitably wind up using our inductive hypothesis (Pk is true) to show that Pk+1 is also
true. But we can't use our hypothesis about Pk unless it somehow appears in Pk+1.
Check out the video to see a working example of the inductive hypothesis where each step is explained. Induction can be a little confusing at first, and it really helps to see it in action.
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.