Problem 26 - College Algebra with Modeling and Visualization

Question

Use a formula to find the sum of the first 20 terms for the arithmetic sequence. $$ a_{1}=-3, d=\frac{2}{3} $$

Step-by-Step Solution

Answer

The sum of the first 20 terms of the sequence is approximately 66.67.

Step 1: Identify the Sequence Parameters

The given arithmetic sequence has the first term \(a_1 = -3\) and a common difference \(d = \frac{2}{3}\). We are asked to find the sum of the first 20 terms of this sequence.

Step 2: Use the Formula for the Sum of an Arithmetic Sequence

The formula to find the sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is given by:\[S_n = \frac{n}{2} (2a_1 + (n-1)d)\]Substitute \(n = 20\), \(a_1 = -3\), and \(d = \frac{2}{3}\) into the formula.

Step 3: Substitute the Given Values

Substitute the values into the sum formula:\[S_{20} = \frac{20}{2} (2(-3) + (20-1)\cdot\frac{2}{3})\]

Step 4: Calculate Inside the Parentheses

Calculate the expression inside the parentheses:\[2(-3) = -6\]\[\text{and } (20-1) \cdot \frac{2}{3} = 19 \cdot \frac{2}{3} = \frac{38}{3}\]So, \[2(-3) + (20-1) \cdot \frac{2}{3} = -6 + \frac{38}{3}\]

Step 5: Combine the Terms

Convert \(-6\) to a fraction with the same denominator to combine:\[-6 = \frac{-18}{3}\]Add \(\frac{-18}{3}\) and \(\frac{38}{3}\):\[\frac{-18}{3} + \frac{38}{3} = \frac{20}{3}\]

Step 6: Compute the Sum

Now substitute back and compute the sum:\[S_{20} = 10 \cdot \frac{20}{3} = \frac{200}{3} = 66.67\]Therefore, the sum of the first 20 terms of the sequence is approximately \(66.67\).

Key Concepts

Sum of Arithmetic Series

The sum of an arithmetic series refers to the total of the first few terms of an arithmetic sequence. An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. Understanding how to find such a sum involves using a specific formula.

The formula for the sum of the first \(n\) terms \(S_n\) in an arithmetic sequence is:

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

Where:

• \(n\) is the number of terms to be added.
• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference between the terms.

To find the sum, you first calculate the expression inside the parentheses, then multiply by \(\frac{n}{2}\). This method ensures you are systematically adding all terms of the sequence from the first to the nth term.

For example, in the provided exercise, calculating the series sum involved substituting \(n = 20\), \(a_1 = -3\), and \(d = \frac{2}{3}\) to discover the sum of the sequence up to the 20th term.

Common Difference

The common difference in an arithmetic sequence is what sets it apart from other types of sequences. It is the fixed number that you add or subtract to each term to get to the next one. This is what makes the sequence consistent and predictable.

Mathematically, if \(a_1\), \(a_2\), \(a_3\), ... are terms in an arithmetic sequence, then the common difference \(d\) can be defined as:

• \(d = a_{n+1} - a_n\)

In the context of the example exercise, the common difference is \(d = \frac{2}{3}\). This means you add \(\frac{2}{3}\) to each term in the sequence to arrive at the next term. Understanding this concept helps in predicting future terms in the sequence without having to laboriously calculate each from scratch.

A clear grasp of the common difference further simplifies the application of the arithmetic sequence formula, as it is an integral part of calculating terms and their sum.

Arithmetic Sequence Formula

The arithmetic sequence formula is central to finding both individual terms and the sum of terms in an arithmetic sequence. This formula helps pinpoint any term in the sequence without needing to write all preceding terms. It's a powerful tool in organizing your work and finding solutions efficiently.

The general formula to find the nth term \(a_n\) in an arithmetic sequence is:

• \[a_n = a_1 + (n-1)d\]

Where:

• \(a_1\) is the first term.
• \(d\) is the common difference.
• \(n\) is the term number you want to find.

By using this formula, you can immediately find the 20th or 30th term in a sequence merely by substituting the values of \(a_1\), \(d\), and your desired term position \(n\).

This function is especially useful in sequences where calculating each term individually is impractical. Furthermore, it serves as a stepping-stone to understanding how to compute the sum of multiple terms using the sum formula.