[answered] MAT 299 Module Eight Homework General: Before beginning thi

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MAT 299 Module Eight Homework

 

General: Before beginning this homework, be sure to read the textbook sections

 

and the material in Module Eight. Type your solutions into this document and be sure to show all steps

 

for arriving at your solution. Just giving a final number may not receive

 

full credit. You may copy and paste mathematical symbols from the statements of

 

the questions into your solution. This document was created using the

 

Arial Unicode font. These problems are proprietary to SNHU COCE, and they may not be

 

posted on any non-SNHU website. The Institutional Release Statement in the course shell gives details

 

about SNHU?s use of systems that compare student submissions to a

 

database of online, SNHU, and other universities? documents. SNHU MAT299 Page 1 of 2 Module Eight Homework 1. What is wrong with this ?proof?? Prove that n! > 2 n.

 

This problem is similar to Exercise 17 in Section 6.1 of your SNHU MAT299

 

textbook.

 

Assume that this is true for n = k. In other words, k! > 2 k.

 

Now

 

(k + 1)!

 

= (k + 1) k!

 

> (k + 1) 2k ? by the induction hypothesis

 

? 2 * 2k? since k ? 1, or k + 1 ? 2

 

= 2k+1.

 

Thus this is true for n = k + 1 and we have proven the theorem.

 

2. Let n ? ? with n ? 1. Prove that 1 + x + x 2 + ? + xn = (1 ? xn+1) / (1 ? x).

 

This problem is similar to Example 6.1.1 and to Exercise 7 in Section 6.1 of

 

your SNHU MAT299 textbook.

 

3. Let n ? ? with n ? 2. Prove that 3n ? n 2n. Hint: If n ? 2, then 3n ? 2n + 2

 

(why?). This problem is similar to Example 6.1.3 and to Exercise 14 in

 

Section 6.1 of your SNHU MAT299 textbook.

 

4. Let n ? ?. Prove that 3 | (4n + 7n + 1). Hint: You can write 7n+1 as (6+1)

 

(7n).

 

This problem is similar to Example 6.1.2 and to Exercise 11 in Section 6.1

 

of your SNHU MAT299 textbook.

 

5. Let n ? ? with n ? 2. Using the multiplication rule for derivatives ? (fg)? =

 

f?g + fg? ? prove that the derivative of xn is nxn?1.

 

This problem should be solved using mathematical induction in Section

 

6.1 of your SNHU MAT299 textbook.

 

6. Let n ? ? with n ? 2. If A1, A2, ?, An are sets, prove that

 

A1 ? A2 ? ? ? An = An ? An?1 ? ? ? A1 where the unions are performed in

 

order from left to right.

 

This problem should be solved using mathematical induction in Section

 

6.1 of your SNHU MAT299 textbook. SNHU MAT299 Page 2 of 2 Module Eight Homework

 

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