Problem 147 - Algebra and Trigonometry Real Mathematics, Real People

Question

Does every finite series whose terms are integers have a finite sum? Explain.

Step-by-Step Solution

Answer

Yes, every finite series of integer terms does have a finite sum. This is due to the closure property of integers under addition, which ensures that the sum of any two integers is always another integer.

Step 1: Understanding the Definitions

A finite series is a sum of a finite number of terms. Here, we have the additional condition that each of these terms is an integer. An integer is a number that can be written without a fractional component and it includes both positive, negative numbers and zero. A finite sum is a sum that ends with a definite number, it is not infinite nor undefined.

Step 2: Properties of Finite Series and Integers

A finite series of integers, like any other finite series, will always have a finite sum. This is a result of the closure property of integers, which states that the sum of any two integers is always an integer. When we add up a finite number of integers, no matter how many there are or how large they may be, we will always obtain an integer.

Step 3: Conclusion

Therefore, it can be concluded that every finite series whose terms are integers does indeed have a finite sum, which is also an integer.

Key Concepts

Integers

Integers are a fundamental concept in mathematics. They include whole numbers without any fractional or decimal components. This category encompasses:

• Positive numbers like 1, 2, 3, etc.
• Negative numbers such as -1, -2, -3, etc.
• Zero, which is neither positive nor negative.

Integers play a vital role in various mathematical operations. They’re foundational in understanding basic arithmetic and complex mathematical theorems. When working with integers, it’s crucial to recognize that they maintain their integrity through operations like addition, subtraction, and multiplication. This means you can add or subtract integers without transforming them into fractions or decimals.

Closure Property

The closure property is an essential concept in understanding why every finite series of integers results in a finite sum. In mathematics, a set is considered to have the closure property if operations performed on elements of the set always yield a result within the same set. For example, when we talk about the integers having a closure property under addition, we mean:

• The sum of any two integers will always be an integer.
• This result remains true regardless of how many integers you add together.

This is particularly important when calculating sums in a finite series. Each time you add two integers, the closure property assures that the new result is still an integer. Thus, a finite sum of integers will always be another integer, maintaining consistency within the set of integers.

Finite Sum

A finite sum refers to the sum of a specific number of terms that stops after reaching a certain count. It contrasts with infinite sums, which continue indefinitely without an endpoint. In finite series, each term is added successively until reaching the last term, resulting in a well-defined total. The sum is finite not only because the series includes a limited number of terms, but also because each term is comprehensible and complete. This is particularly straightforward with integers, given:

• Each number maintains a clear, whole value.
• You end up with a well-defined endpoint, the finite sum.

In essence, a finite sum within a series of integers is a predictable and approachable concept thanks to the integers' nature and their closure under addition. This guarantees that no matter how many terms are included, the sum remains both finite and integer-based.