Suggest a formula for each expression, and prove your conjecture using mathematical induction, n

Suggest a formula for each expression, and prove your...

Key Concepts

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence, where the difference between successive terms is constant. Understanding how to derive and simplify the general formula for an arithmetic series is crucial for efficiently computing the sum of sequences with a uniform additive pattern.

Mathematical Induction

Mathematical induction is a fundamental proof technique used in mathematics to establish that a statement holds true for all natural numbers. It involves proving the base case (usually for n=1) and then showing that if the statement holds for an arbitrary natural number n, it necessarily holds for n+1, thereby validating the statement for every natural number.

Transcript

00:01 I do actually get kind of some enjoyment out of looking for formula for adding together, you know, terms.

00:08 So for instance, i just like to look at, you know, do you see a pattern just for the first term obviously equals two? but if you add the first two terms together, you get six.

00:19 If you add the first three terms together, adding that together, two plus four, six plus six is 12.

00:28 And then the fourth term, and usually you can figure it out, at least i think you can.

00:33 After you get to a certain number term, you know, is there some sort of formula you could think of using the, remember, n is one on this first one to get to two.

00:46 So using n equals one, using n equals two.

00:50 Is there some pattern of getting one term to the next, you know, n equals three to give you 12? and usually a good chunk of times, you can kind of develop something.

01:06 Like you can say, oh, well, i notice that two times one, two times one gives me two.

01:13 And then looking at the next one, oh, two times three.

01:17 So i'm looking at the n is two.

01:19 Two times three gives me six.

01:21 If you're looking at the next one, oh, n is three.

01:24 Well, three times four gives me 12.

01:27 You know, the next one is n equals four.

01:30 Okay, well, looking at the answer of 20, well, if n is four times five.

01:35 Gives me 20 and you might notice a pattern.

01:39 And if you're, you know, on pins and needles trying to figure out what the pattern is, if you still want to figure it out, you can hit pause.

01:48 But the premise is that the formula is n, i just wrote the n over here, and then times n plus one.

01:59 And now we need to prove that that formula is true.

02:03 So how do we do that? we use induction with the base case.

02:08 So is that formula work for the very first term? n equals 1.

02:12 Well, the first term was 2.

02:14 Is that equal to, in my formula, we're putting 1 in for n, 1 plus 1 times 1, is 2 times 1, 2 times 1, 2 times 1, equal to 2? and i would put a checkmark right there.

02:27 Yes, that's true.

02:28 And now what you need to do is the inductive step...

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