Question 3 use the principle of mathematical induction to...

Name: question 3 use the principle of mathematical induction to prove that the following statement is true for all positive integers n 13 23 33 dots n3 fracn2n124 6 question 4 08717
Uploaded: 2021-05-26T23:11:15-08:00
Duration: 12 min 28 s
Channel: Oswaldo Jiménez
Description: question 3 use the principle of mathematical induction to prove that the following statement is true for all positive integers n 13 23 33 dots n3 fracn2n124 6 question 4 08717

QUESTION 3 Use the principle of mathematical induction to prove that the following statement is true for all positive integers n: $1^3 + 2^3 + 3^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$ [6] QUESTION 4

Transcript

00:02 Here we use mathematical induction to prove that sn is true for every positive integer n.

00:09 For that, we follow these steps here.

00:13 Part a we verify s1, and part b we write sk, in part b we write sk plus 1.

00:18 In part b, we write sk plus 1.

00:19 In part b, we write sk plus 1, and in part b we write a conclusion.

00:28 Based on steps a through d, the statement is sk.

00:33 N is 1 cube plus 2 cube plus 3 cube plus up to n -cube equals n squared times n plus 1 square over 4.

00:45 So what we have here on the left is the sum of the cubes of the first n natural numbers.

00:52 And we state that that sum is equal to n squared times n plus 1 squared over 4.

01:00 So in part a we're going to verify as 1.

01:03 When n equals 1, the left side of the sum, of the equality, sorry, is just the first term.

01:15 There is no sum at all, but only the first term, one cube.

01:21 And the right side is 1 square times 1 plus 1 square over 4.

01:29 So the left side is 1, which is 1 cube.

01:33 And on the right side, we have 1, which is 1 square.

01:36 Then times 2 squared over 4, that is 1 equal 1 times 4 over 4, that is 1 equal 4 over 4, that is 1 equal 1.

01:56 That is true.

01:58 And because this is true, we can say that s1 is true.

02:05 And so we have verified the statement for n equals 1, that is 1.

02:12 Now in part p we write the statement as k by replacing n by k in this general statement here above.

02:21 So we get 1 cube plus 2 cube plus 3 cube plus up to k cube equals k squared times k plus 1 square over 4.

02:37 That's the statement for n equals k.

02:41 Now for n equals k plus 1 that is s of k plus 1 is a statement that on the left we have the sum up to the cube of k plus 1 that is 1 q plus 2 q plus 3 q plus we know that we got a stop at k plus 1 cube but we know that before this term we have k cube which is this one here and on the right side we're going to have k plus 1 square times k plus 1 plus 1 square over 4.

03:22 Look that here we have k plus 1 instead of k here.

03:29 And that's the statement k plus 1.

03:33 Now in part d we are going to prove that if s k is true then sk plus 1 is true or any integer.

03:48 Or positive integer k...

Question

QUESTION 3 Use the principle of mathematical induction to prove that the following statement is true for all positive integers n: $1^3 + 2^3 + 3^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$ [6] QUESTION 4

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