Question

We say an integer is even if it can be written in the form 2k, and an integer is odd if it can be written as 2k + 1, where k is an integer. What is the correct expression for an odd integer?

a) n = 2k
b) n = 2k - 1
c) n = 2k + 1
d) n = k/2

Answer 1

The expression for an odd integer is 'n = 2k + 1', where 'k' is an integer. This formula ensures the integer is odd regardless of the value of 'k'.

The correct expression for an odd integer is c) n = 2k + 1

Step-by-step explanation:

The correct expression for an odd integer is c) n = 2k + 1, where k is an integer. When n is an odd number, it can be expressed in the form 2k + 1 because it has a remainder of 1 when divided by 2. This means that no matter what integer value k is assigned, the result will be an odd number. For example, when k is 5, using the expression 2k + 1 gives us 2(5) + 1 = 11, which is an odd integer.

Answer 2

c) n = 2k + 1 This mathematical formula signifies that an odd integer is indeed represented by the expression 2k + 1, where 'k' denotes any integer value. This format allows us to derive any odd integer by substituting various values for 'k' into the equation.

Step-by-step explanation:

The correct expression for an odd integer is n = 2k + 1. In this expression, 'n' represents the odd integer, 'k' is an integer, and '2k' represents an even integer. Adding 1 to an even integer (2k) results in the formation of an odd integer.

To understand this, consider the definition of an odd integer as one that cannot be expressed in the form of 2k (where k is an integer). If we take any integer 'k' and multiply it by 2, the product will always be an even integer (2k). However, by adding 1 to this even integer (2k), we obtain a number that does not conform to the definition of an even integer, hence creating an odd integer.

For instance, let's consider k = 3. If we substitute k = 3 into the expression 2k + 1, it becomes 2(3) + 1 = 6 + 1 = 7, which is an odd integer. This demonstrates that by taking any integer value for 'k' and utilizing the expression 2k + 1, the resulting value will always be an odd integer.

This mathematical formula signifies that an odd integer is indeed represented by the expression 2k + 1, where 'k' denotes any integer value. This format allows us to derive any odd integer by substituting various values for 'k' into the equation.

This approach provides a straightforward and clear representation of odd integers and demonstrates how they differ from even integers. ""