Understand the nature of inductive proof including the use of initial statement, assumption statement, inductive step and conclusion.

Prove that \(\displaystyle\sum_{j=1}^{n} j(j+1) = \frac{n(n+1)(n+2)}{3}\) for all \(n \in \mathbb{Z}^+\) using mathematical induction by completing the steps of the proof as indicated. (a) Initial statement: (1 mark) (b) Assuming the rule is true for \(n = k\), \[1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + k(k+1) = \frac{k(k+1)(k+2)}{3}\] Inductive step: (3 marks) (c) Conclusion: (1 mark)

Consider the following proof by mathematical induction. **Proposition:** For all \( n \in \mathbb{Z}^+ \), the expression \( 7^n - 4^n \) is divisible by 3. Which statement represents the correct formulation of the inductive step that must be proven?

The sum of the first \(n\) cubes is given by \(1^3 + 2^3 + 3^3 + \ldots + n^3 = \left(\frac{n(n+1)}{2}\right)^2\) for all \(n \in \mathbb{Z}^+\). Prove that this rule is true \(\forall n \in \mathbb{Z}^+\) using mathematical induction by completing the steps of the proof as indicated. (a) Initial statement: (1 mark) (b) Assuming the rule is true for \(n = k\), \(1^3 + 2^3 + 3^3 + \ldots + k^3 = \left(\frac{k(k+1)}{2}\right)^2\). Inductive step: (1 mark) (c) Conclusion: (1 mark)

A student is asked to prove by mathematical induction that \(\displaystyle\sum_{j=1}^{n} (3j - 2) = \frac{n(3n-1)}{2}\) for all \(n \in \mathbb{Z}^+\). The student's attempt is shown in the stimulus. (a) Identify which component of a proof by mathematical induction is missing from the student's work. (1 mark) (b) Complete the missing component by showing all necessary working. (2 marks)

Consider a proof by mathematical induction of the proposition \(\sum_{j=1}^{n} j^2 = \frac{n(n+1)(2n+1)}{6}\) for all \(n \in \mathbb{Z}^+\). (a) State the assumption statement that would be used within the proof. (1 mark) (b) Write the expression that represents the left-hand side of the proposition for \(n = k+1\). (1 mark) (c) Explain the purpose of the conclusion statement in a proof by mathematical induction. (1 mark)

Consider a proof by mathematical induction of the proposition \[ \sum_{j=1}^{n} (3j - 2) = \frac{n(3n - 1)}{2} \quad \forall n \in \mathbb{Z}^+ \] (a) State the assumption statement that would be used in the proof. (1 mark) (b) Write the expression that needs to be proven in the inductive step. (1 mark)

Consider the proposition that \(\sum_{j=1}^{n} (3j - 2) = \frac{n(3n-1)}{2}\) for all \(n \in \mathbb{Z}^+\). A student attempts to prove this using mathematical induction and writes four statements shown in the stimulus. Identify which component of an inductive proof each of the following statements represents: (a) Statement P (b) Statement Q (c) Statement R

Prove that \(\displaystyle\sum_{j=1}^{n} j(j+1) = \frac{n(n+1)(n+2)}{3}\) for all \(n \in \mathbb{Z}^+\) using mathematical induction by completing the steps of the proof as indicated. (a) Initial statement: [1 mark] (b) Assuming the rule is true for \(n = k\), \[\sum_{j=1}^{k} j(j+1) = \frac{k(k+1)(k+2)}{3}\] Inductive step: [3 marks] (c) Conclusion: [1 mark]

Consider the proposition that \(\sum_{j=1}^{n} j(j+1) = \frac{n(n+1)(n+2)}{3}\) for all \(n \in \mathbb{Z}^{+}\). Prove this proposition using mathematical induction by completing the following steps: (a) Initial statement: Prove the rule is true for \(n = 1\). (1 mark) (b) Assumption statement: State the assumption for \(n = k\). (1 mark) (c) Inductive step: Assuming the rule is true for \(n = k\), prove it is true for \(n = k+1\). (1 mark) (d) Conclusion: State the conclusion of the proof. (1 mark)

Consider the proposition that \(\displaystyle\sum_{j=1}^{n} j^3 = \left(\frac{n(n+1)}{2}\right)^2\) for all \(n \in \mathbb{Z}^+\). Prove this proposition using mathematical induction by completing the following steps: (a) Initial statement: (1 mark) (b) Assuming the rule is true for \(n = k\), complete the inductive step to prove the rule is true for \(n = k+1\). (3 marks) (c) Conclusion: (1 mark)