Chapter 2.7, Problem 14OE

To determine

To calculate: If $n$ is an integer, is $2n+1$ odd or even? What are the next two off integers greater than $2n+1$ ? What is the next smaller odd integer?

Answer to Problem 14OE

Smaller odd integer than $2n+1$ is $2n-1$ and greater odd integers than $2n+1$ are $2n+3$ and $2n+5$ .

Explanation of Solution

Given information: It is given that $n$ is an integer

Calculation: we can write $2n$ as $2\left(n\right)$ , where $n$ is an integer and we know that if any integer is multiplied by 2 then the resultant is even, so $2\left(n\right)$ is even , that means $2n$ is even.

  $2\left(n\right)+1$ is next integer of $2n$ so $2\left(n\right)+1$ is odd because two consecutive integers cannot be even or odd.

To find smaller odd integer than $2\left(n\right)+1$ , we have to subtract 2 from $2\left(n\right)+1$ , we get

  $2n+1-2=2n-1$

To find greater odd integer than $2n+1$ , we have to add 2 to $2n+1$ , we get

  $2n+1+2=2n+3$

To find the greater odd integer than $2n+3$ , we have to add 2 to $2n+3$ , we get

  $2n+3+2=2n+5$

Conclusion: Smaller odd integer than $2n+1$ is $2n-1$ and greater odd integers than $2n+1$ are

  $2n+3$ and $2n+5$ .