Problem 2 - Algebra 2

Question

Write the related series for each finite sequence. Then evaluate each series. $$ -5,-15,-25,-35,-45 $$

Step-by-Step Solution

Answer

The related series can be written as: \[ S = \frac{5}{2} * (-5 - 45) \]. Evaluating this gives us: \[ S = -125 \].

Step 1: Identify common difference

First, need to find the common difference of the arithmetic sequence. This can be done by subtracting any number in the sequence by the one before it. In this example, if we subtract -15 (the second number) by -5 (the first number) we get a common difference of -10.

Step 2: Write the related Series

Next, write the related arithmetic series. An arithmetic series can be written as: \[ S = \frac{n}{2} * (a + l) \], where 'n' is the number of terms (5 in this case), 'a' is the first term (-5), and 'l' is the last term (-45). By substituting these values into the formula, the arithmetic series can be denoted.

Step 3: Evaluate the Series

By substituting our values into the formula, we get: \[ S = \frac{5}{2} * (-5 - 45) \]. Once calculated, the sum of the series S can be found.

Key Concepts

Finite Sequence

A finite sequence is a set of numbers listed in a specific order with a definite number of terms. Unlike an infinite sequence which never ends, a finite sequence stops at a certain point. In our example sequence, we have the numbers

• -5
• -15
• -25
• -35
• -45

These numbers are ordered and the sequence ends at -45, making it a finite sequence with 5 terms. This definitiveness is what categorizes it distinctly. Each number in the sequence is called a term. The order is essential because the sequence considers both which numbers appear and also their order of appearance.

Common Difference

The common difference is a key concept in arithmetic sequences. It indicates how each term progresses to the next by a fixed amount. In an arithmetic sequence, this difference remains constant across all consecutive terms. To find it:

• Pick any two successive terms in the sequence.
• Perform subtraction: next term minus previous term.

For instance, if you take -15 and subtract -5, the common difference is calculated as \(-15 - (-5) = -10\).
This tells us that each term in the sequence decreases by 10 from the one before it.
Identifying the common difference is crucial for understanding the pattern and for further calculations, like evaluating a series.

Evaluate Series

Evaluating a series involves calculating the sum of all terms within a sequence. Specifically, for an arithmetic sequence, we use the formula:\[ S = \frac{n}{2} \cdot (a + l) \] where:

• \(n\) is the total number of terms.
• \(a\) is the first term in the sequence.
• \(l\) is the last term.

In this example, \( n = 5 \), \( a = -5 \), and \( l = -45 \). Plugging these into our formula gives:\[ S = \frac{5}{2} \cdot (-5 + (-45)) \] Calculating this results in the sum of the series. Each evaluation provides the total value summed from all sequence terms, expressing the overall accumulation between the start and end of the sequence.

Arithmetic Sequence

An arithmetic sequence is a type of sequence characterized by a constant difference between any two consecutive terms. This common difference is what defines the arithmetic nature of the sequence. In our given sequence:

• -5
• -15
• -25
• -35
• -45

We observe that each term decreases by 10, which is the common difference.
This means if you start at -5 and repeatedly add -10 (or subtract 10), you'll enumerate through the sequence. Understanding arithmetic sequences is beneficial for modeling consistent incremental changes or patterns of addition and subtraction, which frequently occur in real-world scenarios.