Prove each formula by induction, if possible. 2+5+8+…+(3 n-1)=(n(3 n+1))/(2)

Name: Problem 15, Chapter 8: College Algebra
Uploaded: 2020-07-07T20:05:55-08:00
Duration: 6 min 54 s
Channel: Kevin Harmer
Description: Prove each formula by induction, if possible. 2+5+8+…+(3 n-1)=(n(3 n+1))/(2)

Prove each formula by induction, if possible. 2+5+8+…+(3...

Key Concepts

Inductive Step

The component of an induction proof where, using the inductive hypothesis, one demonstrates that the statement true for n implies the statement is true for n+1, thus linking each case to the next.

Mathematical Induction

A proof technique used to establish that a statement holds for all natural numbers. It involves proving the base case and then showing that if the statement is true for an arbitrary integer n, it must also be true for n+1.

Base Case

The initial step in an induction proof where the proposition is verified for the first value in the sequence, often n = 1, to serve as the foundational truth for further steps.

Inductive Hypothesis

The step in the process where one assumes that the statement holds for some arbitrary natural number n. This assumption is then utilized to establish the truth of the statement for n+1.

Arithmetic Series

A sequence of numbers in which the difference between consecutive terms is constant. The sum of an arithmetic series can be determined using known formulas and serves as the framework for verifying summation identities like the one given.

Transcript

00:01 To prove this using induction, we first have to look at the base case at n equals 1.

00:06 So we have on our left -hand side, our first term is 2, and that's going to be equal to n, which is 1, times 3n, so 3 times 1, plus 1, all divided by 2.

00:24 So we solve this, it's pretty easy.

00:28 We have the coefficient of 1, and then we have 3 plus 1 is 4 divided by 2, so we're left with 2.

00:35 And we can see pretty easily that the base case works for this induction point.

00:44 Now we have to look at the inductive case.

00:47 So given the n is equal to k, we know that 2 plus 5 plus 8 plus a whole bunch of other numbers with the same pattern is going to be equal to 3n minus 1.

01:10 And n in this case is going to be k.

01:15 And that's equal to n or k times 3k, minus 1 divided by 2.

01:28 So we go about solving this by adding on the next term.

01:39 So mathematically, this is shown by adding on, which is represented in red, 3 times k plus 1 minus 1 term to the end of our left hand side, our sum of all these terms.

02:02 Now notice this is just the equivalent for our induction step with n equal to k plus 1, k plus 1 instead of n is equal to k.

02:19 So we continue that's equal to 3 times 3k plus 1, all divided by 2.

02:35 And then we have a plus 3 times 3k plus 1.

02:49 And that second 3 is a mistake.

02:52 That shouldn't be there.

02:53 So now we keep going...

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