To prove: If $n$ is a positive integer, then $n2+2n$ is either odd or a multiple of 4.

Given Information:

A positive integer $n$ .

Concept Used:

If $n$ is odd then it must be of the form $n=2k+1$ , where $k$ is a positive integer.

If $n$ is even then it must be of the form $n=2k$ , where $k$ is a positive integer.

Proof:

Let $n$ be a positive integer.

Then $n$ can be either even or odd.

Case 1: Let $n$ be odd.

If $n$ is odd then it must be of the form $n=2k+1$ , where $k$ is a positive integer.

Substitute $n=2k+1$ into $n2+2n$ and simplify as follows.

$n2+2n=2k+12+22k+1=4k2+1+4k+4k+2=4k2+8k+2+1=22k2+4k+1+1=2m+1 2k2+4k+1=m$

Thus, if $n$ is odd then $n2+2n$ is also odd.

Case 2: Let $n$ be even.

If $n$ is even then it must be of the form $n=2k$ , where $k$ is a positive integer.

Substitute $n=2k$ into $n2+2n$ and simplify as follows.

$n2+2n=2k2+22k=4k2+4k=4k2+k=4m k2+k=m$

Thus, if $n$ is even then $n2+2n$ is a multiple of 4.

Therefore, from case 1 and 2 it follows that if $n$ is a positive integer, then $n2+2n$ is either odd or a multiple of 4.

Chapter 1.1, Problem 52E

Chapter 1.1, Problem 54E

Chapter 1 Solutions

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Prob. 7QR Ch. 1.4 - Prob. 8QR Ch. 1.4 - Prob. 9QR Ch. 1.4 - Prob. 10QR Ch. 1.4 - In Exercises 14 find formulas for the functions...Ch. 1.4 - In Exercises 14 find formulas for the functions...Ch. 1.4 - Prob. 3E Ch. 1.4 - Prob. 4E Ch. 1.4 - Prob. 5E Ch. 1.4 - Prob. 6E Ch. 1.4 - Prob. 7E Ch. 1.4 - Prob. 8E Ch. 1.4 - Prob. 9E Ch. 1.4 - Prob. 10E Ch. 1.4 - Prob. 11E Ch. 1.4 - Prob. 12E Ch. 1.4 - Prob. 13E Ch. 1.4 - Prob. 14E Ch. 1.4 - Prob. 15E Ch. 1.4 - Prob. 16E Ch. 1.4 - Prob. 17E Ch. 1.4 - Prob. 18E Ch. 1.4 - Prob. 19E Ch. 1.4 - Prob. 20E Ch. 1.4 - Prob. 21E Ch. 1.4 - Prob. 22E Ch. 1.4 - Prob. 23E Ch. 1.4 - Prob. 24E Ch. 1.4 - Prob. 25E Ch. 1.4 - Prob. 26E Ch. 1.4 - Prob. 27E Ch. 1.4 - Prob. 28E Ch. 1.4 - Prob. 29E Ch. 1.4 - Prob. 30E Ch. 1.4 - Prob. 31E Ch. 1.4 - Prob. 32E Ch. 1.4 - Prob. 33E Ch. 1.4 - Prob. 34E Ch. 1.4 - Prob. 35E Ch. 1.4 - Prob. 36E Ch. 1.4 - Prob. 37E Ch. 1.4 - Prob. 38E Ch. 1.4 - Prob. 39E Ch. 1.4 - Prob. 40E Ch. 1.4 - Prob. 41E Ch. 1.4 - Prob. 42E Ch. 1.4 - Prob. 43E Ch. 1.4 - 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Solution Summary: The author explains that if n is a positive integer, it must be of the form n=2k+1.

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