Use mathematical induction in Exercises 3-17 to prove...

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis. Let $P(n)$ be the statement that $1^{3}+2^{3}+\cdots+n^{3}=(n(n+$ 1$) / 2 )^{2}$ for the positive integer $n .$ a) What is the statement $P(1) ?$ b) Show that $P(1)$ is true, completing the basis step of the proof of $P(n)$ for all positive integers $n .$ c) What is the inductive hypothesis of a proof that $P(n)$ is true for all positive integers $n$ ? d) What do you need to prove in the inductive step of a proof that $P(n)$ is true all positive integers $n ?$ e) Complete the inductive step of a proof that $P(n)$ is true for all positive integers $n,$ identifying where you use the inductive hypothesis. f) Explain why these steps show that this formula is true whenever $n$ is a positive integer.

Use mathematical induction in Exercises $3-17$ to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
Let $P(n)$ be the statement that $1^{3}+2^{3}+\cdots+n^{3}=(n(n+$ 1$) / 2 )^{2}$ for the positive integer $n .$
a) What is the statement $P(1) ?$
b) Show that $P(1)$ is true, completing the basis step of the proof of $P(n)$ for all positive integers $n .$
c) What is the inductive hypothesis of a proof that $P(n)$ is true for all positive integers $n$ ?
d) What do you need to prove in the inductive step of a proof that $P(n)$ is true all positive integers $n ?$
e) Complete the inductive step of a proof that $P(n)$ is true for all positive integers $n,$ identifying where you use the inductive hypothesis.
f) Explain why these steps show that this formula is true whenever $n$ is a positive integer.

Key Concepts

Inductive Step

The inductive step is the part of the proof where, using the inductive hypothesis that the statement holds for an arbitrary integer k, one shows that the statement must also hold for k+1. This step is essential in propagating the truth from the base case through all subsequent natural numbers, thereby completing the overall proof by induction.

Summation Formulae

Summation formulae are expressions that represent the sum of a sequence of numbers, often involving powers such as cubes or squares. Proving such formulae using induction not only confirms their correctness for each natural number but also provides insight into the structure and patterns underlying numerical summations.

Inductive Hypothesis

The inductive hypothesis is the assumption made for some arbitrary positive integer k that the statement or formula holds true. In an induction proof, this hypothesis is used in the subsequent step to demonstrate that if the statement is valid for one number, it must also be valid for the next number, k+1.

Base Case

The base case is the initial step in a proof by induction where the statement is verified for the first number in the sequence (usually n = 1). This step is crucial as it establishes the starting point of the mathematical induction process, ensuring that the statement holds at its inception.

Mathematical Induction

Mathematical induction is a proof technique used to establish that a given statement is true for all positive integers. It involves two main steps: verifying the base case and then proving that if the statement is true for an arbitrary integer k (the inductive hypothesis), it must also be true for k+1. This method leverages the well-ordered property of the natural numbers to extend the truth from one case to the next.

Transcript

00:01 In this question, we are asked to prove by induction the statement that said the sum of integers cube from 1 to n equals this term, n times n plus 1 over 2 square.

00:20 So we will call this statement p of n for n as some positive integers.

00:28 By using induction, we have 2.

00:31 Step to follows first the basic step is the first case that the statement holds true and in this case is when n equals to one that's not always the case that it has to start with one just that in this case one is the basic step other case may start from to five 11 depending on question so don't don't stuck on this n equal to one always okay so p of one we just substitute n as one right so on this side there will be sum of just one term and on this side we we have this number is to one square so it's one right that means p1 said 1 equals to so it's true.

01:37 Then we are clear this basic step.

01:39 Next, the inductive step.

01:41 We assume that if the statement is true for some n that we want, so it's greater or equal to one, some positive integer.

01:57 We want to show that n plus 1 is true as well.

02:00 So this hasn't happened yet.

02:04 We have to show it.

02:05 And how to show it? we can use p n as the starting point because it's like an assume statement so we have it as facts as information now consider the sum from 1 to n plus 1 cube this first n term you you can see that we can use inductive hypothesis right because it is n is up to n so p of n work.

02:42 That means we can change it into this term and the last term stay the same...

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