Problem 21 - Algebra 2

Question

Find \(S_{n}\) for each arithmetic series described. $$ a_{1}=12, d=\frac{1}{3}, n=13 $$

Step-by-Step Solution

Answer

The sum of the series \(S_{13}\) is 182.

Step 1: Identify the Formula for the Sum of an Arithmetic Series

To find the sum of an arithmetic series, we use the formula for the sum of the first \(n\) terms: \[S_n = \frac{n}{2}(2a_1 + (n-1)d)\] where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the number of terms.

Step 2: Substitute Values into the Formula

Substitute \(a_1 = 12\), \(d = \frac{1}{3}\), and \(n = 13\) into the formula: \[S_{13} = \frac{13}{2}(2(12) + (13-1)\times\frac{1}{3})\] which simplifies to \[S_{13} = \frac{13}{2}(24 + 12\times\frac{1}{3})\].

Step 3: Simplify the Expression Inside the Parentheses

Calculate the expression inside the parentheses: First calculate \(12 \times \frac{1}{3}\), which equals 4. Now substitute back: \[S_{13} = \frac{13}{2}(24 + 4)\] which simplifies to \[S_{13} = \frac{13}{2}(28)\].

Step 4: Calculate the Sum

Now, compute \(\frac{13}{2} \times 28\). First, calculate \(13 \times 28 = 364\). Then divide by 2: \[S_{13} = \frac{364}{2} = 182\].

Key Concepts

Sum of an Arithmetic Series

One of the fascinating aspects of arithmetic series is how we can find the sum of its terms quickly and efficiently. When we have a list of numbers, or terms, that increase or decrease by a consistent amount, this list forms an arithmetic series. Instead of adding each term individually, which can be time-consuming, we use a formula that allows us to find the sum of the first \( n \) terms easily. This is known as the "Sum of an Arithmetic Series Formula."

The formula for finding the sum of an arithmetic series is:

• \( S_n = \frac{n}{2}(2a_1 + (n-1)d) \)

In this formula:

• \( S_n \) is the sum of the first \( n \) terms,
• \( a_1 \) is the first term of the series,
• \( d \) is the common difference between terms,
• \( n \) is the total number of terms.

Through this formula, you can plug in your values and directly calculate the sum without having to add each term separately. This saves time and reduces the risk of computational errors.

Arithmetic Series Formula

The arithmetic series formula is central to finding sums efficiently. This formula leverages the consistent progression between numbers in an arithmetic series. Given that each term in the series is derived from the one before it by adding a common difference \( d \), the formula allows you to step through these values with ease.

A quick overview of utilizing this formula involves these steps:

• Identify the first term \( a_1 \).
• Determine the common difference \( d \).
• Know the number of terms \( n \) you wish to sum.

Once these components are in place, substituting them into the standard arithmetic series formula \( S_n = \frac{n}{2}(2a_1 + (n-1)d) \) allows you to calculate the desired sum. This powerful tool simplifies the addition of multiple terms, providing clarity and ease to what could otherwise be a tedious task.

Common Difference in Series

The common difference in an arithmetic series is the core component that defines its structure. It's the fixed amount that each term in the series increases or decreases by. Understanding the common difference is crucial as it lets us predict every term in the series just from knowing the first term and the difference.

To find the common difference \( d \):

• Take any term in the series and subtract the previous term from it.

For example, in a series where each term adds \( \frac{1}{3} \) to the previous one, \( \frac{1}{3} \) becomes the common difference. If the first term \( a_1 = 12 \) and we know the common difference, we can extend the series or apply it within the arithmetic series sum formula to find sums efficiently.

Recognizing the common difference helps in anticipating the growth or reduction of the series and plays a central role in calculating not only the terms themselves but also their total sum.