Problem 39 - Algebra 2

Question

Evaluate each series to the given term. $$ 0.17+0.13+0.09+0.05+\ldots ; 12 $$

Step-by-Step Solution

Answer

-0.6

Step 1: Identify the first term, common difference and number of terms

The first term \(a\) of the series is 0.17. The common difference \(d\) is found by subtracting successive terms, which gives -0.04. The series has 12 terms, so \(n = 12\).

Step 2: Find the last term

Calculate the last term \(l\) of the series using the formula: \(l = a + (n - 1) * d\). Substitute the given values to get \(l = 0.17 + (12 - 1) * -0.04 = -0.27\).

Step 3: Calculate the sum of the series

Use the formula for the sum of an arithmetic series: \(S = n/2 * (a + l)\). Substitute the calculated and given values to get \(S = 12/2 * (0.17 - 0.27) = -0.6\).

Key Concepts

Common Difference

Understanding the common difference is crucial for identifying an arithmetic series. It's the constant amount added (or subtracted) from one term to the next in the series. In an arithmetic sequence, each term after the first can be found by adding this common difference to the previous term.
For example, in the series given:

• The first term is 0.17, followed by subsequent terms 0.13, 0.09, 0.05, and so on.
• To find the common difference, subtract any term from the term before it. Observing from the series, subtracting 0.13 from 0.17 gives -0.04. Similarly, subtracting 0.09 from 0.13 also gives -0.04.
• This constant difference of -0.04 indicates that each term is 0.04 less than the previous one, characterizing it as a decreasing series.

Recognizing this step is essential as it provides the means to calculate further terms and eventually the sum of the series.

Sum of Series

The sum of an arithmetic series is calculated using a specific formula that takes into account the number of terms, the first term, and the last term of the series. For any arithmetic series, the sum can be calculated using the formula:

• \[ S = \frac{n}{2} \times (a + l) \]

Where:

• \( S \) is the sum of the series,
• \( n \) is the number of terms,
• \( a \) is the first term, and
• \( l \) is the last term.

In our exercise, we have 12 terms in total. The first term is 0.17, and by using the discussed common difference, we calculated the last term to be -0.27.
The sum of this series is then calculated as:

• \[ S = \frac{12}{2} \times (0.17 + (-0.27)) = -0.6 \]

This sum tells us the combined total when all terms are added together, which is particularly useful in various applications of mathematics.

First Term

The first term of an arithmetic series is the initial term from which the series begins. It serves as the foundation upon which the series is built. Knowing the first term is pivotal as it shapes the progression of the entire sequence.
In the arithmetic series provided in the exercise, the first term is 0.17. This sets the starting point of the series, and is a crucial value in determining other attributes of the sequence, such as the common difference and further terms.
The first term is used prominently in the formula to find the sum of the series. It signifies the beginning of the pattern and is added together with the last term to determine the sum. Understanding this helps in grasping the overall structure and calculation of arithmetic series.

• First term \( (a) = 0.17 \)
This concept reemphasizes the necessity of examining the first value keenly as it influences all subsequent calculations involving the arithmetic sequence.