Question

Evaluate the sums in Problems 5-12 using the formula for the sum of an arithmetic series. $$ \sum_{n=1}^{10}(5-4 n) $$

Short answer

$$ Answer: The sum of the arithmetic series is \(-170\).

Step-by-step solution

Identify the number of terms

The given sum is from \(n=1\) to \(n=10\), so there are 10 terms in the series.

Identify the first term

To find the first term, we plug \(n=1\) into the expression: $$ a_1 = 5 - 4(1) = 1. $$

Identify the last term

To find the last term, we plug \(n=10\) into the expression: $$ a_n = 5 - 4(10) = 5-40 = -35. $$

Use the arithmetic series formula

Now that we have our values for \(n\), \(a_1\), and \(a_n\), we can plug these into the arithmetic series formula: $$ S = \dfrac{n}{2} (a_1 + a_n) = \dfrac{10}{2} (1 - 35) = -170. $$

Final Answer

Therefore, the sum of the arithmetic series is \(-170\).

Key concepts

Sum of Series

When we talk about the 'sum of series,' particularly in arithmetic sequences, we are referring to the total of all the numbers in a sequence. This is calculated by adding each term from the first to the last. It's crucial to use a formula for efficiency, especially when dealing with a large number of terms or complex numbers. By employing a formula, we can swiftly find the sum without individually adding each number. This approach saves time and eliminates the potential for errors in manual calculations.

• In an arithmetic series, each term increases or decreases by a constant amount.
• Using a systematic approach allows you to solve problems more efficiently.
• The sum provides a quick snapshot of the total influence or effect of the entire series.

This method allows you not only to tackle school problems easily but also aids in understanding patterns in real-life sequences.

First Term

The first term of an arithmetic series is essential because it establishes the starting point from which all other terms are derived. In our exercise, the first term, denoted as \(a_1\), is determined by setting \(n=1\) in the series expression. For a sequence described by \(5 - 4n\), substituting \(n = 1\) provides: \[a_1 = 5 - 4 imes 1 = 1\]

• Understanding the first term helps predict the behavior of subsequent terms.
• It is used directly in the formula to calculate the sum of the series.
• Ensures an accurate sequence setup before proceeding with calculations.

This term serves as the foundation upon which the rest of the series is systematically constructed.

Last Term

In arithmetic series, identifying the last term is just as crucial as finding the first. The last term, often referred to as \(a_n\), is calculated by plugging the final value of \(n\) (here, 10) into the expression describing the series. For the series \(5 - 4n\), we find:\[a_n = 5 - 4 imes 10 = -35\]

• The last term helps define the boundary of the series.
• It is a critical component in determining the series sum using the formula.
• This term signifies the conclusion of the sequence, marking its final value.

With the last term known, you can effectively apply the sum formula and understand the entire range covered by the series.

Arithmetic Series Formula

The arithmetic series formula is a straightforward tool that calculates the sum of a series without needing to add each term manually. This formula is:\[S = \dfrac{n}{2} (a_1 + a_n)\]where:

• \(S\) is the sum of the series.
• \(n\) represents the total number of terms in the series.
• \(a_1\) is the first term.
• \(a_n\) is the last term.

To calculate the sum, you add the first and last terms, multiply by the number of terms, and divide by 2. For our problem:\[S = \dfrac{10}{2} (1 - 35) = -170\]This formula is not only efficient but reliable, helping quickly determine sums across various mathematical contexts. Using the formula correctly ensures that calculations are both accurate and easy to perform.