Problem 28 - Algebra 2

Question

Tell whether each list is a sequence or a series. Then tell whether it is finite or infinite. $$ -0.5-0.25-0.125-\ldots $$

Step-by-Step Solution

Answer

The given list represents an infinite sequence.

Step 1: Identify if the list is a sequence or a series

A sequence is a list of numbers in a specific order. A series is the sum of the terms of a sequence. Here, the list given is neither being added nor subtracted but just listed one after the other. Therefore, this list represents a sequence.

Step 2: Identify whether the sequence is finite or infinite

A finite sequence has a definite number of terms, whereas an infinite sequence continues indefinitely. Here the sequence is followed by an ellipsis (\ldots), which suggests that it goes on indefinitely. So it's an infinite sequence.

Key Concepts

finite and infinite

When studying sequences, it is essential to discern between finite and infinite sequences. A **finite sequence** is one that has a limited number of terms. For example, if a sequence lists numbers such as 1, 2, 3, and stops at 10, it is finite—it has an ending point.
In contrast, an **infinite sequence** does not have a definite ending point. It will go on forever, without stopping. When a sequence is followed by an ellipsis ("..."), it indicates that the sequence is infinite. This symbol tells us the sequence continues indefinitely, like the sequence in the exercise: -0.5, -0.25, -0.125, ... In this context, we don't know the last term, because there isn't one; it keeps going.

Understanding the difference between finite and infinite is crucial when you're analyzing sequences. It helps you know whether you're dealing with a set amount of data or possibilities that extend endlessly.

mathematical sequences

Mathematical sequences are fundamental concepts in math that involve lists of numbers arranged in a particular order. Each number in that list is referred to as a **term**, and the terms are often generated based on a specific rule or formula.
Sequences can be classified in various ways, but broadly, they connect under these categories:

• **Arithmetic sequences** where each term is formed by adding a constant to the previous one. Example: 2, 5, 8 where the constant is 3.
• **Geometric sequences** in which each term is found by multiplying the previous term by a fixed, non-zero number. For instance, 2, 4, 8 has a constant multiplier of 2.

In the given exercise, -0.5, -0.25, -0.125 is an example of a geometric sequence where each term is half of the previous one.
Next, we'll look at series, which is how these sequences can be transformed into something different.

identifying sequences and series

Understanding the difference between sequences and series is vital in mathematical studies. A **sequence** is simply a list of numbers in a specific order. Think of it as a pattern with a rule, whether linear, quadratic, or based on another mathematical operation that lists numbers one after another.
On the other hand, a **series** is the sum of the terms of a sequence. So if you were to take a sequence and add the terms together, you would form a series. For example, say your sequence is 1, 2, 3; the series would be 1 + 2 + 3.

In the exercise, we see a list of numbers: -0.5, -0.25, -0.125, and they are presented just as a sequence because we aren't adding them together. The lack of summation indicates it remains a sequence, not transforming into a series.
Recognizing whether you're working with a sequence or series is key in solving mathematical problems since it dictates the approach and method you will use.