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tags: mth325-f22
---

# Content Skill Standard Quiz 1

:::info
This quiz contains *new questions* for **Content Skill Standards P.1 and P.2**.

**Instructions**

* Work only the problems that you **need to work** and **feel ready to work**.
* Do your work on separate pages with **each Content Skill Standard on its own separate page**. *Please do not put work for multiple Standards on the same page.*
* Make sure to consult the [Specifications for Student Work in MTH 325](https://hackmd.io/lD6oyEN5RdiUi_wdg-rkZg) document before starting on your work, so you're clear on what is expected and what constitutes a "successful" attempt.
* Also, make sure to carefully read the **Success Criteria** below each problem to know exactly what is expected from that problem.
* When you are done: **Scan** each Learning Target to a clear, legible, black-and-white PDF and **upload the PDF** to the designated folder on Blackboard. Please *do not just take a picture with your camera* --- use a scanning app to create a PDF, and upload the PDF. You can also type up your work and save to a PDF if you want; or use a notes app on a tablet to handwrite your work and save that to a PDF.
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## Content Skill Standard P.1

:::warning
★ P.1: Given a statement to prove with mathematical induction, I can identify the predicate, state and prove the base case, state the inductive hypothesis, and outline the rest of the proof.
:::

Consider the statement:

>For all positive integers $n$, $1 + 5 + 9 + 13 + \cdots + (4n-3) = n(2n-1)$.

1. State the predicate involved in this statement.
2. State and prove the base case for a proof by induction of this statement.
3. State the inductive hypothesis for a proof by induction of this statement.
4. Outline the remainder of a proof by induction for this statement by clearly stating what you would prove, and giving at least one plausible idea for proving it.

**Success criteria:** The predicate must be clearly stated; the base case must be correctly stated and proven; and the inductive hypothesis must be correctly stated in the context of the problem. The outline for the remainder of the proof must clearly state what you are going to prove, and the "plausible idea" must be some specific step you might take that makes sense in the context of the problem.

## Content Skill Standard P.2

:::warning
★ P.2: Given a conditional statement, I can state the assumptions and conclusions for a direct proof, proof by contrapositive, and proof by contradiction.
:::

Consider the statement:

>For every integer $n$, if $n^3 + 5$ is odd, then $n$ is even.

1. Clearly state what you would assume and what you would prove, if proving this statement with a *direct proof*.
2. Clearly state what you would assume and what you would prove, if proving this statement with a *proof by contrapositive*.
3. Clearly state *all* the assumptions you would make, if proving this statement *by contradiction*.

**Success criteria:** All parts of each of the three items here are correctly and clearly stated.