Problem 3 - 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. $$ 2+4+6+8+10+12 $$

Step-by-Step Solution

Answer

The sum of the series is 42.

Step 1: Identify the sequence components

Recognize that the series is an arithmetic sequence where the first term \(a_1\) is 2 and the common difference \(d\) is 2. The last term \(a_n\) of the series is 12.

Step 2: Determine the number of terms

Use the formula for the nth-term of an arithmetic sequence \(a_n = a_1 + (n-1) \cdot d\) to find the number of terms \(n\). Set \(a_n = 12\): \(12 = 2 + (n-1) \cdot 2\). Solving for \(n\), we get \(n=6\).

Step 3: Apply the formula for the sum of an arithmetic series

Use the sum formula for an arithmetic series: \(S_n = \frac{n}{2} (a_1 + a_n)\). Substitute \(n=6\), \(a_1=2\), and \(a_n=12\) to find the sum: \(S_6 = \frac{6}{2} (2 + 12)\).

Step 4: Calculate the sum

Simplify the equation from the previous step: \(S_6 = 3 \cdot 14 = 42\). Thus, the sum of the series is 42.

Key Concepts

Understanding Arithmetic Sequences

An arithmetic sequence is a list of numbers where each term after the first is generated by adding a fixed number, known as the common difference, to the previous term. In our exercise, the numbers 2, 4, 6, 8, 10, and 12 form an arithmetic sequence. Each term is obtained by adding 2 to the one before it.

• The first term is often denoted as \(a_1\).
• The common difference is represented by \(d\).
• The formula to find the nth-term is \(a_n = a_1 + (n-1)d\).

In this sequence, \(a_1 = 2\) and \(d = 2\). To find the sixth term, you use \(a_6 = 2 + (6-1) \cdot 2 = 12\). The list is thus 2, 4, 6, 8, 10, and it ends at 12, as calculated.

Finding the Partial Sum

A partial sum is the sum of the first few terms of a sequence, in this case, the first six terms of our arithmetic sequence. Knowing that this sequence is finite, we can compute the total using the partial sum of an arithmetic series. Recognizing the first (\(a_1\)) and last term (\(a_n\)) of the portion you want to sum up is crucial. Here, you sum from 2 to 12.

• Start from the first term.
• Include all the terms up to 12 which gives you a clear finite stopping point.

This is called the partial sum because it doesn't necessarily include every possible term in the arithmetic sequence, just the ones up to \(a_6 = 12\).

Identifying the Common Difference

The common difference in an arithmetic sequence is the constant amount that each term increases or decreases by in order to get to the next term. Understanding this value is fundamental, as it can help you determine any term in the sequence. In our example, the common difference \(d\) is 2.

• If \(d\) is positive, the sequence is increasing.
• If \(d\) is negative, the sequence is decreasing.

Here, 2 is added to 2 to get 4, to 4 to get 6, and so forth. This constant addition is what defines the uniform nature of an arithmetic sequence.

Using the Sum Formula

To find the total of the first few terms in an arithmetic sequence, the sum formula for an arithmetic series is used. This formula, \(S_n = \frac{n}{2} (a_1 + a_n)\), provides a straightforward method to find the sum when you know \(n\), which is how many terms you are considering, \(a_1\), the first term, and \(a_n\), the last term of the sequence.

• \(n\) represents the number of terms.
• \(a_1\) is the starting term of the sequence.
• \(a_n\) is the final term included in the sum.

In our exercise, using \(a_1 = 2\), \(a_n = 12\), and \(n = 6\): \[S_6 = \frac{6}{2} (2 + 12) = 3 \times 14 = 42\]Thus, the sum of the sequence is 42. This formula simplifies the work by avoiding adding up each number individually, especially useful for longer sequences.