Use the Principle of Mathematical Induction to show that the given statement is true for all nat

Use the Principle of Mathematical Induction to show that the...

Key Concepts

Inductive Step

The inductive step is the part of the proof where one uses the inductive hypothesis to demonstrate that if the statement holds for a natural number n, then it must necessarily hold for n + 1. This step is crucial as it ensures that the truth of the statement 'propagates' from one natural number to the next, thereby proving the statement for all natural numbers.

Inductive Hypothesis

The inductive hypothesis involves assuming that the statement under consideration holds for an arbitrary natural number n. This assumption is a critical step that sets up the framework for proving that the statement must also hold for the next natural number, n + 1.

Base Case

The base case is the first step in a proof by induction, where the statement is verified for the initial natural number (usually n = 1). Establishing the truth of the base case is essential because it provides the starting point of the induction process.

Mathematical Induction

Mathematical induction is a proof technique used to establish that a statement holds for all natural numbers. The method involves verifying the statement for an initial case (often n = 1) and then proving that if the statement is true for an arbitrary natural number n, it must also be true for n + 1, thereby covering the entire set of natural numbers.

Transcript

00:01 Okay, so we want to use mathematical induction to show that the following is true.

00:05 So our first step is to show that n is equal to 1 is true.

00:09 So let's write out n is equal to 1, but that means we have 3, or we can say 1 plus 2, correct? we're using this equation here, but that's equal to 3, and is that equal to our right -hand side? let's write out our right -hand side.

00:25 This is 1 -half times 1, and then we have 1 plus 5.

00:31 That's 6 over 2, which is equal to 3.

00:34 So we see that n is equal to 1 holds.

00:37 And now let's assume that n is equal to k is true.

00:41 That is we have 3 plus 4 plus 5 all the way up to k plus 1, or sorry, that k plus 2, is equal to 1⁄2 k times k plus 5.

00:56 Okay, and now we want to show that n is equal to k plus 1.

01:03 Is true.

01:04 So let's first write that out, assuming that it is true, we would get the equation above with k plus 1 instead.

01:16 So we would get k plus 3 here, and then we would have one half times k plus 1 and then times k plus 1 plus 5, which is 6.

01:28 Okay, so let's work with our left hand side.

01:32 So we can rewrite the following as 3 plus 4 plus 5, all the way up to k plus 2.

01:42 And then we also have, that's our term before, k plus 3.

01:46 And why did we want to do this? well, we know that this portion is our n is equal to k.

01:53 And we assume that this was true...

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