Like with programming, it is good for readability to declare your variables near the top, and the introduction is the top of the inductive proof. Example introduction: We will use induction to prove that k < 2k for k = 1,2,3... We will denote this inequality as P(k) = k < 2k. We will start by proving the base case. Base caseInduction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $ 1. Induction proof of a Recurrence Relation? 0. Induction proof with binomials. 2. Prove the Binomial Theorem using Induction. 0. Proof by Induction of an inequality with a sum. 1. Am I properly using induction (specifically the induction hypothesis)? 4.There’s a lot that goes into buying a home, from finding a real estate agent to researching neighborhoods to visiting open houses — and then there’s the financial side of things. F...The 8 Major Parts of a Proof by Induction: First state what proposition you are going to prove. Precede the statement by Proposition, Theorem, Lemma, …Say you're given an array of zeroes and ones. You cannot say that "since 0 indicates a boolean 'false' and the first item in the array is zero, therefore the rest of the array will also correspond to 'false.'". Proof by induction is only good for problems that work according to strong mathematical rules. 2.20 Nov 2020 ... How did we arrive at this guess? Well, we noted that for every doubling of n, we increased the value of T(n) by three. In order to count the ...Feb 15, 2022 · Proof by induction: strong form. Example 1. Example 2. One of the most powerful methods of proof — and one of the most difficult to wrap your head around — is called mathematical induction, or just “induction" for short. I like to call it “proof by recursion," because this is exactly what it is. The Principle of Mathematical Induction is used to prove mathematical statements suppose we have to prove a statement P (n) then the steps applied are, Step 1: Prove P (k) is true for k =1. Step 2: Let P (k) is true for all k in N and k > 1. Step 3: Prove P (k+1) is true using basic mathematical properties.Proof by Induction Steps. The steps to use a proof by induction or mathematical induction proof are: Prove the base case. (In other words, show that the property is true for a specific value of n ...Proof by Induction. We proved above that 0 is a neutral element for + on the left using a simple partial evaluation argument. The fact that it is also a neutral element on the right ... Theorem plus_0_r_firsttry : ∀n: nat, n + 0 = n. ... cannot be proved in the same simple way.Question. Prove by contradiction that the equation 2x 3 + 6x + 1 = 0 has no integer roots.. ️Answer/Explanation. Ans: METHOD 1 (rearranging the equation) assume there exists some α∈ Z such that 2α 3 + 6α + 1 = 0. Note: Award M1 for equivalent statements such as ‘assume that α is an integer root of \(2\alpha ^{3} + 6\alpha +1 = 0′.\) Condone the use of …Jun 15, 2007 · Send. An induction proof of a formula consists of three parts a Show the formula is true for b Assume the formula is true for c Using b show the formula is true for For c the usual strategy for a summation is to manipulate into the form Induction is a method for checking a result discovering the result may be hard. 1 Proofs by Induction. Induction is a method for proving statements that have the form: 8n : P (n), where n ranges over the positive integers. It consists of two steps. First, you …What is induction in calculus? In calculus, induction is a method of proving that a statement is true for all values of a variable within a certain range. This is done by showing that the statement is true for the first term in the range, and then using the principle of mathematical induction to show that it is also true for all subsequent terms. What are the steps for proof by induction with series? STEP 1: The basic step. Show the result is true for the base case; This is normally n = 1 or 0 but it could be any integer . For example: To prove is true for all integers n ≥ 1 you would first need to show it is true for n = 1: ; STEP 2: The assumption step. Assume the result is true for n = k for some integer kInduction. 🔗. Mathematical induction is a proof technique, not unlike direct proof or proof by contradiction or combinatorial proof. 3 In other words, induction is a style of argument we use to convince ourselves and others that a mathematical statement is always true. Many mathematical statements can be proved by simply explaining what they ... How to prove summation formulas by using Mathematical Induction.Support: https://www.patreon.com/ProfessorLeonardProfessor Leonard …Sep 30, 2023 · Proof by Induction. Proof by induction is a technique used in discrete mathematics to prove universal generalizations. A universal generalization is a claim which says that every element in some series has some property. For example, the following is a universal generalization: For any integer n ≥ 3, 2^n > 2n. State and prove the inductive step. The inductive step in a proof by induction is to show that for all choices of k, if P ( k) is true, then P ( k + 1) is true. Typically, you'd prove this by assuming P ( k) and then proving P ( k + 1). We recommend specifically writing out both what the assumption P ( k) means and what you're going to prove ...3.1 Mathematical induction You have probably seen proofs by induction over the natural numbers, called mathematicalinduction. In such proofs, we typically want to prove that some property Pholds for all natural numbers, that is, 8n2N:P(n). A proof by induction works by first proving that P(0) holds, and then proving for all m2N, if P(m) then P ... Nov 21, 2023 · Proof by Induction Steps. The steps to use a proof by induction or mathematical induction proof are: Prove the base case. (In other words, show that the property is true for a specific value of n ... 3. It is useful to think of induction proofs as an "outline" for an infinite length proof. In particular, what you a providing is a way to write a proof for any particular n. For example, say you've proven 1 + 2 +... + n = n ( n + 1) / 2 by induction. We can think of this as giving me a 'program' to write a proof for, say, n = 6 or n = 100000 ...1 Proofs by Induction. Induction is a method for proving statements that have the form: 8n : P (n), where n ranges over the positive integers. It consists of two steps. First, you prove that P (1) is true. This is called the basis of the proof. May 20, 2022 · Process of Proof by Induction. There are two types of induction: regular and strong. The steps start the same but vary at the end. Here are the steps. In mathematics, we start with a statement of our assumptions and intent: Let \(p(n), \forall n \geq n_0, \, n, \, n_0 \in \mathbb{Z_+}\) be a statement. Proof by mathematical induction. An example of the application of mathematical induction in the simplest case is the proof that the sum of the first n odd positive integers is n 2 —that is, that (1.) 1 + 3 + 5 +⋯+ (2n − 1) = n 2 for every positive integer n.TOPICS. Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index New in MathWorldsingle path through inductive proofs: the ext step" may need creativity. We will meet proofs by induction involving linear algebra, polynomial algebra, calculus, and exponents. In each proof, nd the statement depending on a positive integer. Check how, in the inductive step, the inductive hypothesis is used. Some results below are about Proof by Induction. Induction is a method of proof usually used to prove statements about positive whole numbers (the natural numbers). Induction has three steps: The base case is where the statement is shown to be true for a specific number. Usually this is a small number like 1.Learn how to prove statements by induction, a fundamental proof technique that is useful for proving that a statement is true for all positive integers n. See the formula, the …A sample problem demonstrating how to use mathematical proof by induction to prove recursive formulas.In this video, I explain the proof by induction method and show 3 examples of induction proofs! :DInstagram:https://www.instagram.com/braingainzofficialGetting a good night’s sleep is essential for our overall well-being and productivity. Unfortunately, many of us struggle with noise disturbances that can disrupt our sleep pattern...What is induction in calculus? In calculus, induction is a method of proving that a statement is true for all values of a variable within a certain range. This is done by showing that the statement is true for the first term in the range, and then using the principle of mathematical induction to show that it is also true for all subsequent terms. Proof by inductions questions, answers and fully worked solutionsThis section briefly introduces three commonly used proof techniques: deduction, or direct proof; proof by contradiction and. proof by mathematical induction. In general, a direct proof is just a “logical explanation”. A direct proof is sometimes referred to as an argument by deduction. This is simply an argument in terms of logic.Revision Village - Voted #1 IB Math Resource! New Curriculum 2021-2027. This video covers Proof by Mathematical Induction. Part of the IB Mathematics Analysi...Mar 20, 2022 · Let n n and k k be non-negative integers with n ≥ k n ≥ k. Then. ∑i=kn (i k) = (n + 1 k + 1) ∑ i = k n ( i k) = ( n + 1 k + 1) Proof. This page titled 3.8: Proofs by Induction is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Mitchel T. Keller & William T. Trotter via source content that was edited to ... Proof by Mathematical Induction - How to do a Mathematical Induction Proof ( Example 2 ) In this tutorial I show how to do a proof by mathematical induction.Join this channel to …Apr 17, 2022 · List the first 10 Lucas numbers and the first ten Fibonacci numbers and then prove each of the following propositions. The Second Principle of Mathematical Induction may be needed to prove some of these propositions. (a) For each natural number n, Ln = 2fn + 1 − fn. (b) For each n ∈ N with n ≥ 2, 5fn = Ln − 1 + Ln + 1. What is induction in calculus? In calculus, induction is a method of proving that a statement is true for all values of a variable within a certain range. This is done by showing that the statement is true for the first term in the range, and then using the principle of mathematical induction to show that it is also true for all subsequent terms. Nov 27, 2023 · Proof by Induction. Induction is a method of proof usually used to prove statements about positive whole numbers (the natural numbers). Induction has three steps: The base case is where the statement is shown to be true for a specific number. Usually this is a small number like 1. 2 Feb 2014 ... Proof by Induction ... In order to prove a mathematical identity, one needs to show that the identity is valid for all the values in the desired ...First, multiply both sides of the inequality by \ (xy\), which is a positive real number since \ (x > 0\) and \ (y > 0\). Then, subtract \ (2xy\) from both sides of this inequality and finally, factor the left side of the resulting inequality. Explain why the last inequality you obtained leads to a contradiction.What are the steps for proof by induction with matrices? · State the result is true · Explain in words why the result is true · It must include: If true for n ...Lecture 2: Induction Viewing videos requires an internet connection Description: An introduction to proof techniques, covering proof by contradiction and induction, with an emphasis on the inductive techniques used in proof by induction.Induction is also useful in any level of mathematics that has an emphasis on proof. Induction problems can be found anywhere from the Power Round of the ARML up ...No matter if you’re opening a bank account or filling out legal documents, there may come a time when you need to establish proof of residency. There are several ways of achieving ...Proof by induction is a robust and diverse method of mathematical proof used when the result or final expression is already known. In AQA A-Level Further Mathematics, it is involved only in proving sums of series, divisibility, and powers of matrices. The four-stage process is always as follows: Base case: Prove the result is true for = 1 (or 0).Using AM-GM inequality (which can be proved by induction on the number of terms), Equality holds iff 12 =22 = … = n2 1 2 = 2 2 = … = n 2, which means equality does not hold for n > 1 n > 1. which can be proved by induction on n n. which can also be proved by induction on n n. Taking the n n th power on both sides (which preserves order as ...The key step of any induction proof is to relate the case of \(n=k+1\) to a problem with a smaller size (hence, with a smaller value in \(n\)). Imagine you want to send a letter that requires a \((k+1)\)-cent postage, and you can use only 4-cent and 9-cent stamps. You could first put down a 4-cent stamp.Proof by induction is a robust and diverse method of mathematical proof used when the result or final expression is already known. In AQA A-Level Further Mathematics, it is involved only in proving sums of series, divisibility, and powers of matrices. The four-stage process is always as follows: Base case: Prove the result is true for = 1 (or 0).In today’s fast-paced and ever-changing business landscape, it is crucial for brands to stay ahead of the curve and anticipate what comes next. This is where future-proofing your b...Are you tired of ordering pizza delivery every time you crave a delicious slice? Why not try making your own pizza at home? With the right techniques, you can create a mouthwaterin...Lecture 2: Induction Viewing videos requires an internet connection Description: An introduction to proof techniques, covering proof by contradiction and induction, with an emphasis on the inductive techniques used in proof by induction.The Principle of Mathematical Induction is used to prove mathematical statements suppose we have to prove a statement P (n) then the steps applied are, Step 1: Prove P (k) is true for k =1. Step 2: Let P (k) is true for all k in N and k > 1. Step 3: Prove P (k+1) is true using basic mathematical properties.single path through inductive proofs: the ext step" may need creativity. We will meet proofs by induction involving linear algebra, polynomial algebra, calculus, and exponents. In each proof, nd the statement depending on a positive integer. Check how, in the inductive step, the inductive hypothesis is used. Some results below are about Jan 12, 2015 · Then, the book moves on to standard proof techniques: direct proof, proof by contrapositive and contradiction, proving existence and uniqueness, constructive proof, proof by induction, and others. These techniques will be useful in more advanced mathematics courses, as well as courses in statistics, computers science, and other areas. Apr 16, 2018 at 14:55. 4. The assumption of the inductive hypothesis is valid because you have proven (in the first part of the proof by induction, the base case) that the statement P P holds for n =n0 n = n 0. So you can think of it this way: initially, you only know that P(n0) P …2.1 Mathematical induction You have probably seen proofs by induction over the natural numbers, called mathematicalinduction. In such proofs, we typically want to prove that some property Pholds for all natural numbers, that is, 8n2N:P(n). A proof by induction works by first proving that P(0) holds, and then proving for all m2N, if P(m) then P ...What are the steps for proof by induction with sequences? STEP 1: The basic step. Show the result is true for the base case. If the recursive relation formula for the next term involves the previous two terms then you need to show the position-to-term formula works the first two given terms which will be given as part of the definition of the sequence 4 Feb 2013 ... Comments111 · Induction: Series & Algebraic Identities (1 of 4) · Induction Divisibility · Introduction to Proof by Mathematical Induction.by the induction hypothesis. = 11(5m) + 66 − 6. by expanding the bracket. = 5(11m) + 60 = 5(11m + 12) since both parts of the formula have a common factor of 5. As 11m + 12 is an integer we have that 11k+1 − 6 is divisible by 5, so P (k + 1) is correct. Hence by mathematical induction P (n) is correct for all positive integers n.Proof by Induction A proof by induction is a way to use the principle of mathematical induction to show that some result is true for all natural numbers n. In a proof by induction, there are three steps: Prove that P(0) is true. – This is called the basis or the base case. Prove that if P(k) is true, then P(k+1) is true. – This is called the inductive step.Domino Fall Down 2. With this metaphor, proof by induction consists in two steps. First, we need to make sure that the first domino will fall. This corresponds to the basic case. Then, we need to check whether all dominoes are perfectly alined, such that every domino will make the next one fall.Learn how to prove the sum of all positive integers up to and including n by induction, a method of mathematical proof that establishes a statement for all natural numbers. Watch a video tutorial and see the formula, steps, and examples with solutions. Viewed 840 times. 2. I have to make the following proof: n ∑ k = 1k(n k) = n2n − 1. Base case, n = 1: 1 ∑ k = 1k(1 k) = 1 = 1 ⋅ 20 = 1 Inductive Hypothesis: for int p = n p ∑ k = 1k(p k) = p2p − 1. Inductive Step; here is where I am having some trouble....This explains the need for a general proof which covers all values of n. Mathematical induction is one way of doing this. 1.2 What is proof by induction? One way of thinking about mathematical induction is to regard the statement we are trying to prove as not one proposition, but a whole sequence of propositions, one for each n. The trick used ...An inductive load pulls a large amount of current when first energized, then settles down to a full-load running current after a few seconds or cycles. When switched, inductive loa...The Well-Ordering Principle guarantees that the proof by contradiction works by exhibiting a least element of S S. If some n ∈N n ∈ N makes the predicate P P false, then there is a least such . As s ≥ 2 s ≥ 2, the natural number before s s, namely s − 1 s − 1, must make P P true. – Berrick Caleb Fillmore. Apr 19, 2015 at 7:10.Lecture 2: Induction Viewing videos requires an internet connection Description: An introduction to proof techniques, covering proof by contradiction and induction, with an emphasis on the inductive techniques used in proof by induction.Deer can be a beautiful addition to any garden, but they can also be a nuisance. If you’re looking to keep deer away from your garden, it’s important to choose the right plants. He...Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/algebra-home/alg …Proof by induction. In mathematics, we use induction to prove mathematical statements involving integers. There are two types of induction: regular …by José Pablo Iriarte in Uncanny Magazine Issue Forty | 6296 Words. Paulie rushes out the elevator doors the moment they part, only to skid to a halt at the ...MadAsMaths :: Mathematics ResourcesProof by Induction Suppose that you want to prove that some property P(n) holds of all natural numbers. To do so: Prove that P(0) is true. – This is called the basis or the base case. Prove that for all n ∈ ℕ, that if P(n) is true, then P(n + 1) is true as well. – This is called the inductive step. – P(n) is called the inductive hypothesis.Jun 28, 2023 · Proof by induction. In mathematics, we use induction to prove mathematical statements involving integers. There are two types of induction: regular and strong. The steps start the same but vary at the end. Here are the steps. In mathematics, we start with a statement of our assumptions and intent: MadAsMaths :: Mathematics ResourcesProve the base case holds true. As before, the first step in any induction proof is to prove that the base case holds true. In this case, we will use 2. Since 2 is a prime number (only divisible by itself and 1), we can conclude the base case holds true. 4.In today’s rapidly evolving job market, it is crucial to stay ahead of the curve and continuously upskill yourself. One way to achieve this is by taking advantage of the numerous f...
You have probably seen proofs by induction over the natural numbers, called mathematicalinduction. In such proofs, we typically want to prove that some property Pholds for all natural numbers, that is, 8n2N:P(n). A proof by induction works by first proving that P(0) holds, and then proving for all m2N, if P(m) then P(m+1).. Good days
2 Feb 2014 ... Proof by Induction ... In order to prove a mathematical identity, one needs to show that the identity is valid for all the values in the desired ...1 Proofs by Induction. Induction is a method for proving statements that have the form: 8n : P (n), where n ranges over the positive integers. It consists of two steps. First, you …Theorem 1.3. 2 - Generalized Principle of Mathematical Induction. Let n 0 ∈ N and for each natural n ≥ n 0, suppose that P ( n) denotes a proposition which is either true or false. Let A = { n ∈ N: P ( n) is true }. Suppose the following two conditions hold: n 0 ∈ A. For each k ∈ N, k ≥ n 0, if k ∈ A, then k + 1 ∈ A.Induction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $ 1. Induction proof of a Recurrence Relation? 0. Induction proof with binomials. 2. Prove the Binomial Theorem using Induction. 0. Proof by Induction of an inequality with a sum. 1. Am I properly using induction (specifically the induction hypothesis)? 4.Learn how to prove the sum of all positive integers up to and including n by induction, a method of mathematical proof that establishes a statement for all natural numbers. …Proofs by transfinite induction typically distinguish three cases: when n is a minimal element, i.e. there is no element smaller than n; when n has a direct predecessor, i.e. the set of elements which are smaller than n has a largest element; when n has no direct predecessor, i.e. n is a so-called ... 21 Jan 2014 ... A Sample Proof Using Mathematical Induction (playing with LaTeX). It's been a long time since I used LaTeX regularly, and I discovered that I ...Mar 26, 2012 · Here you are shown how to prove by mathematical induction the sum of the series for r squared. ∑r²YOUTUBE CHANNEL at https://www.youtube.com/ExamSolutionsEXA... MadAsMaths :: Mathematics ResourcesDeductive research aims to test an existing theory while inductive research aims to generate new theories from observed data. Deductive research works from the more general to the ...A proof by induction involves two steps: Proving that IF the above formula is true for any particular value of n, let's say n=k, then it must automatically follow that it isrue for k+1 too. Since (k+1) is another particular value, the same argument shows the formula is therefore true for k+2. "By induction" we can therefore reason that it will ...Basically, an induction proof isn't a proof, it's a blueprint for building a proof in a finite number of steps. The induction hypothesis is a function that takes a proof and returns a proof. Let's say you want to prove P(5), but you've already proven P(1), and you have a function IH that takes P(n) to P(n+1) regardless of the value of n. Then ...Proof by Mathematical Induction - How to do a Mathematical Induction Proof ( Example 2 ) In this tutorial I show how to do a proof by mathematical induction.Join this channel to …TOPICS. Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index New in MathWorldThe reason why this is called "strong induction" is that we use more statements in the inductive hypothesis. Let's write what we've learned till now a bit more formally. Proof by strong induction. Step 1. Demonstrate the base case: This is where you verify that \(P(k_0)\) is true. In most cases, \(k_0=1.\) Step 2. Prove the inductive step:5.1.3 A Template for Induction Proofs. The proof of equation (5.1) was relatively simple, but even the most complicated induction proof follows exactly the same template. There are five components: 1. State that the proof uses induction. This immediately conveys the overall structure of the proof, which helps your reader follow your argument. 2. Inductive Step: ∀ k, P ( k) → P ( k + 1) is true. Then P ( n) is true for all positive integers n. This definition uses n = 1 as the base case, but the induction argument can shifted and started at any integer n = a. In this case one needs to prove the base case P ( a) is true along with the inductive step. Learn how to prove the sum of all positive integers up to and including n by induction, a method of mathematical proof that establishes a statement for all natural numbers. Watch a video tutorial and see the formula, steps, and examples with solutions. The overall form of the proof is basically similar, and of course this is no accident: Coq has been designed so that its induction tactic generates the same sub-goals, in the same order, as the bullet points that a mathematician would write. But there are significant differences of detail: the formal proof is much more explicit in some ways (e.g., the use of reflexivity) ….
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Nordam appliance | The Principle of Mathematical Induction is used to prove mathematical statements suppose we have to prove a statement P (n) then the steps applied are, Step 1: Prove P (k) is true for k =1. Step 2: Let P (k) is true for all k in N and k > 1. Step 3: Prove P (k+1) is true using basic mathematical properties.Proof by Induction Counterexamples Appendix Answer Key Symbols Used in this Book Glossary The beauty of induction is that it allows a theorem to be proven …The Induction Principle: Let P(n) be a statement which depends on n = 1,2,3,···. Then P(n) is true for all n if: • P(1) is true (the base case). • Prove ......
Coco songs | It is defined to be the summation of your chosen integer and all preceding integers (ending at 1). S (N) = n + (n-1) + ...+ 2 + 1; is the first equation written backwards, the reason for this is it becomes easier to see the pattern. 2 (S (N)) = (n+1)n occurs when you add the corresponding pieces of the first and second S (N).prove by induction product of 1 - 1/k^2 with k from 2 to n = (n + 1)/(2 n) for n>1. Prove divisibility by induction: using induction, prove 9^n-1 is divisible by 4 assuming n>0. induction 3 divides n^3 - 7 n + 3. Derive a proof by induction of …...
Cheap flights from houston to atlanta | Proof by Induction Counterexamples Appendix Answer Key Symbols Used in this Book Glossary The beauty of induction is that it allows a theorem to be proven …Deer can be a beautiful addition to any garden, but they can also be a nuisance. If you’re looking to keep deer away from your garden, it’s important to choose the right plants. He......
Descarga musica gratis | How to prove summation formulas by using Mathematical Induction.Support: https://www.patreon.com/ProfessorLeonardProfessor Leonard Merch: https://professor-l...The reason why this is called "strong induction" is that we use more statements in the inductive hypothesis. Let's write what we've learned till now a bit more formally. Proof by strong induction. Step 1. Demonstrate the base case: This is where you verify that \(P(k_0)\) is true. In most cases, \(k_0=1.\) Step 2. Prove the inductive step: ...
How to cure a zombie villager | In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction . Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as ...The Principle of Mathematical Induction is used to prove mathematical statements suppose we have to prove a statement P (n) then the steps applied are, Step 1: Prove P (k) is true for k =1. Step 2: Let P (k) is true for all k in N and k > 1. Step 3: Prove P (k+1) is true using basic mathematical properties.Proof by induction · Language · Watch · Edit. Redirect page. Redirect to: Mathematical induction....
Indoor walking tracks near me | What are proofs? Proofs are used to show that mathematical theorems are true beyond doubt. Similarly, we face theorems that we have to prove in automaton theory. There are different types of proofs such as direct, indirect, deductive, inductive, divisibility proofs, and many others. Proof by induction. The axiom of proof by induction states that:Algebra (all content) 20 units · 412 skills. Unit 1 Introduction to algebra. Unit 2 Solving basic equations & inequalities (one variable, linear) Unit 3 Linear equations, functions, & graphs. Unit 4 Sequences. Unit 5 System of equations. Unit 6 Two-variable inequalities. Unit 7 Functions. Unit 8 Absolute value equations, functions, & inequalities. I've recently been trying to tackle proofs by induction. I'm having a hard time applying my knowledge of how induction works to other types of problems (divisibility, inequalities, etc). I've been checking out the other induction questions on this website, but they either move too fast or don't explain their reasoning behind their steps enough ......