Problem 30 - Algebra 2

Question

Tell whether each list is a sequence or a series. Then tell whether it is finite or infinite. $$ 2.3+4.6+9.2+18.4 $$

Step-by-Step Solution

Answer

The list is a finite series.

Step 1: Identify the Mathematical Structure

Examine the list of numbers to determine whether they are terms of a sequence or terms being added together as in a series.

Step 2: Sequence or Series Determination

In the given list, numbers are being added together. This indicates that it is a series because a series is a sum of terms of a sequence.

Step 3: Determine the Finiteness

Since there is a clear end to the list of numbers with no indication that they continue indefinitely, it is a finite series.

Key Concepts

Sequence vs Series

Understanding the difference between a sequence and a series is crucial for students tackling mathematical exercises. A sequence is essentially a set of numbers arranged in a specific order. It is an ordered list of numbers where each number is called a term. For example, the list (2, 4, 6, 8) represents a sequence of even numbers.

A series, on the other hand, is related to a sequence, yet distinct. It is the sum of the terms of a sequence. To move from a sequence to a series, you would essentially add the terms together. If you were to take the earlier sequence (2, 4, 6, 8) and perform addition, you'd have a series: 2 + 4 + 6 + 8.

When analyzing a list of numbers, pinpoint whether you are simply observing them as they are listed (sequence) or adding them up (series). If the list you're examining consists of terms that are being added, then you're definitely dealing with a series.

Arithmetic Series

Dive into the concept of an arithmetic series, you are engaging with a series that stems from an arithmetic sequence. Recall that an arithmetic sequence is one where each term after the first is obtained by adding a constant difference to the previous term. This difference is often referred to as the 'common difference'.

For example, consider the sequence 3, 5, 7, 9, where each term increases by 2. This is an arithmetic sequence with a common difference of 2. Turning this into a series, we get the arithmetic series 3 + 5 + 7 + 9. It represents the sum of the arithmetic sequence terms.

Arithmetic series have unique properties and formulas that allow us to calculate the sum efficiently, especially if the series is finite and we know how many terms it includes. This reduces a potentially time-consuming addition process to a simple application of a formula.

Sum of a Series

Calculating the sum of a series is a fundamental skill in mathematics. For a finite arithmetic series, there's a quick way to find the sum without having to add each term individually. This magical method involves using a formula:\[ S_n = \frac{n}{2}(a_1 + a_n) \]In this formula, \( S_n \) represents the sum of the first \( n \) terms of the series, \( a_1 \) is the first term, \( a_n \) is the last term, and \( n \) is the number of terms in the series.

Let's apply this formula using the example of the arithmetic series 3 + 5 + 7 + 9 again. Here, \( a_1 \) is 3, \( a_4 \) is 9, and there are 4 terms (\( n = 4 \)). Plugging the numbers into the formula gives \( S_4 = 2(3 + 9) = 24 \). This is the sum of the series, which matches what you'd get if you added each term together. This formula is an invaluable shortcut for finding the sum of any finite arithmetic series quickly.