Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-8,\) find the sum of each series using the formula for the partial sum of an arithmetic series. Be sure to show your work. $$ 0 i+4 i+8 i+12 i+16 i+20 i $$

Step-by-Step Solution

Answer

The sum of the series is \(60i\).

Step 1: Identify the Series

First, identify the series given in the problem. The series is: \(0i, 4i, 8i, 12i, 16i, 20i\). This is an arithmetic series with a common difference.

Step 2: Find the First Term and Common Difference

Identify the first term and the common difference of the series. The first term \(a_1 = 0i\). The second term \(a_2 = 4i\). Thus, the common difference \(d = a_2 - a_1 = 4i\).

Step 3: Determine the Number of Terms

Count the number of terms in the series to find \(n\). The series has the terms \(0i, 4i, 8i, 12i, 16i, 20i\), which count up to 6 terms (n=6).

Step 4: Use the Formula for the Sum of an Arithmetic Series

Use the formula for the sum of an arithmetic series: \[ S_n = \frac{n}{2} \times (a_1 + a_n) \]where \(S_n\) is the sum, \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term.

Step 5: Calculate the Last Term

The last term \(a_n\) in the series is \(20i\).

Step 6: Calculate the Sum

Substitute the values into the sum formula:\[ S_6 = \frac{6}{2} \times (0i + 20i) \]Perform the calculations:\[ S_6 = 3 \times 20i = 60i \]This gives the sum of the series.

Key Concepts

Sum of Arithmetic Series

When dealing with arithmetic series, finding the sum can be simplified using a specific formula. This is known as the partial sum formula. Let's explore the idea behind it.

• Imagine you're looking at a series like the one given: \(0i, 4i, 8i, 12i, 16i, 20i\).
• You could manually add up each term, but using the sum formula will save you time and effort.
• The formula for the sum of the first \(n\) terms (\(S_n\)) of an arithmetic series is:
  \[ S_n = \frac{n}{2} \times (a_1 + a_n) \] where \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term.

By understanding this formula, you can easily compute the sum of any arithmetic series without adding each term individually.

Common Difference

In arithmetic series, one of the key features is the common difference, denoted as \(d\). This is what makes the series arithmetic.

• The common difference is simply the amount you add to each term to get the next term.
• For example, starting at \(0i\), adding \(4i\) gets you to \(4i\), and then adding another \(4i\) reaches \(8i\).
• Mathematically, \(d\) is calculated by finding the difference between any two consecutive terms:
  \(d = a_2 - a_1\).

The common difference helps define the unique nature of the series and is pivotal in calculations.

Partial Sum Formula

The partial sum formula is a powerful tool that makes calculating the sum of an arithmetic series straightforward. Let's break down how this works.

• The formula \(S_n = \frac{n}{2} \times (a_1 + a_n)\) helps find the sum of the first \(n\) terms quickly.
• Using this formula reduces the complexity as you only need the first term \(a_1\) and the last term \(a_n\), instead of every single term in between.

You start by determining these terms and then applying them within the formula. The sum is calculated by multiplying the average of the first and last term by the number of terms, divided by two. Such methodologies save time and also reduce errors in calculation.

Number of Terms in a Series

A crucial step when working with the partial sum formula is determining the number of terms in the series, represented as \(n\).

• First, list out or identify all terms within the series.
• Counting the number of terms gives you \(n\), which is essential for calculating the sum.
• For instance, in our example, \(0i, 4i, 8i, 12i, 16i, 20i\), counting makes it clear that there are 6 terms, so \(n = 6\).

Correctly counting the terms ensures that the application of the partial sum formula results in the accurate computation of the series' sum.