Problem 3 - College Algebra

Question

Is the ordered set of even numbers an infinite sequence? What about the ordered set of odd numbers? Explain why or why not.

Step-by-Step Solution

Answer

Both sets, the even and odd numbers, form infinite sequences because they can continue indefinitely without end.

Step 1: Define the sequence of even numbers

An even number is any integer that can be divided by 2 without leaving a remainder, such as 0, 2, 4, 6, 8, etc. Mathematically, even numbers can be expressed as the sequence: \( a_n = 2n \) where \( n \) is an integer starting from 0.

Step 2: Determine if the sequence of even numbers is infinite

The formula \( a_n = 2n \) for even numbers can produce a new even number for every whole number \( n \). Since there is no upper limit to how high \( n \) can go, the sequence of even numbers is infinite.

Step 3: Define the sequence of odd numbers

An odd number is any integer that gives a remainder of 1 when divided by 2, such as 1, 3, 5, 7, 9, etc. Mathematically, odd numbers can be expressed as the sequence: \( b_n = 2n + 1 \) where \( n \) is an integer starting from 0.

Step 4: Determine if the sequence of odd numbers is infinite

The formula \( b_n = 2n + 1 \) for odd numbers can produce a new odd number for every whole number \( n \). Similar to the even numbers, since there is no limit to the value of \( n \), the sequence of odd numbers is also infinite.

Step 5: Conclusion

Both sequences, the ordered set of even numbers and the ordered set of odd numbers, are infinite because you can keep generating numbers indefinitely by increasing \( n \) without any upper limit.

Key Concepts

Even Numbers

Even numbers are integers that can be exactly divided by 2. This means no remainder is left after the division.
For example, numbers like 0, 2, 4, 6, and 8 are even numbers because they can be expressed as multiples of 2.
This gives us a simple mathematical representation: \( a_n = 2n \), where \ n\ is a non-negative integer starting from zero. This means:

• When \( n = 0 \), \( a_n = 2(0) = 0 \)
• When \( n = 1 \), \( a_n = 2(1) = 2 \)
• When \( n = 2 \), \( a_n = 2(2) = 4 \)

By using this formula, we can keep generating even numbers without stopping. Since \( n \) can keep increasing indefinitely, this sequence has no endpoint, making it infinite.

Odd Numbers

Odd numbers, on the other hand, are integers that leave a remainder of 1 when divided by 2.
Examples include numbers like 1, 3, 5, 7, and 9. These numbers can be generated by the expression \( b_n = 2n + 1 \), where \( n \) is again a non-negative integer starting from zero.

• When \( n = 0 \), \( b_n = 2(0) + 1 = 1 \)
• When \( n = 1 \), \( b_n = 2(1) + 1 = 3 \)
• When \( n = 2 \), \( b_n = 2(2) + 1 = 5 \)

Just like even numbers, these odd numbers also form an infinite sequence. The absence of a highest \( n \) ensures that the sequence of odd numbers can continue forever.

Integer Division

Integer division refers to dividing one integer by another and finding the quotient without considering any remainder.
When dividing by 2, this concept helps in understanding what makes a number "even" or "odd". For even numbers, integer division by 2 results in a whole number.
For example:

• 6 divided by 2 equals 3, which is exact without a remainder, indicating 6 is even.
• 8 divided by 2 equals 4, again exact with no remainder, proving 8 is even.

Conversely, when an integer like 7 is divided by 2, the quotient is 3 with a remainder of 1.
This remainder is what classifies 7 as an odd number. Integer division is a key mathematical operation that underpins the sequence formulas for both even and odd numbers, solidifying their definitions in arithmetic.