Problem 58 - A Graphical Approach to College Algebra

Question

Use a formula to find the sum of each arithmetic series. The first 50 terms of the series \(a_{n}=1-3 n\)

Step-by-Step Solution

Answer

The sum of the first 50 terms is -3775.

Step 1: Understand the Series

The given series is defined by the formula \(a_n = 1 - 3n\). We need to find the sum of the first 50 terms of this arithmetic sequence.

Step 2: Identify the First Term

The first term of the series \(a_1\) can be found by substituting \(n = 1\) into the formula: \(a_1 = 1 - 3(1) = -2\). So, the first term \(a_1\) is \(-2\).

Step 3: Identify the Common Difference

The common difference \(d\) in an arithmetic series is the difference between successive terms. For this series, using \(a_n = 1 - 3n\), the common difference \(d\) is \(-3\) because each subsequent term decreases by 3.

Step 4: Identify the Last Term

Using the formula for the nth term \(a_n = 1 - 3n\), and knowing we want the first 50 terms, find the last term \(a_{50} = 1 - 3(50) = -149\).

Step 5: Use the Sum Formula for an Arithmetic Series

The sum \(S_n\) of the first \(n\) terms of an arithmetic series is given by the formula \(S_n = \frac{n}{2} (a_1 + a_n)\). For this series, \(n = 50\), \(a_1 = -2\), and \(a_{50} = -149\).

Step 6: Calculate the Sum

Substitute into the sum formula: \[S_{50} = \frac{50}{2} (-2 + (-149)) = 25(-151) = -3775\]. The sum of the first 50 terms is \(-3775\).

Key Concepts

Sum of Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is generated by adding a constant, known as the common difference, to the previous term. The sum of an arithmetic sequence, also known as an arithmetic series, can be calculated using a straightforward formula. This formula helps in finding the sum of a specified number of terms without having to add each term manually.

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

• \( S_n = \frac{n}{2} (a_1 + a_n) \)

Here, \( S_n \) is the total sum, \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the nth term of the sequence. This formula stems from the idea of averaging the first and the last terms and multiplying by the number of terms, providing both an efficient and reliable way to find the sum.

Common Difference

The common difference in an arithmetic sequence is the consistent difference between consecutive terms. Recognizing the common difference is crucial, as it determines the pattern of the sequence.

For a sequence defined by the formula \( a_n = 1 - 3n \), the common difference \( d \) can be understood by simplifying the expression: each term reduces by 3 as \( n \) increases by 1. Thus, \( d = -3 \).

This negative sign means that the sequence is decreasing, illustrating how the common difference impacts the overall progression of the sequence.

First Term

In an arithmetic sequence, the first term is crucial as it serves as the starting point for the entire sequence. For the series defined by \( a_n = 1 - 3n \), the first term \( a_1 \) is obtained by substituting \( n = 1 \) into the sequence's formula.

By calculation, \( a_1 = 1 - 3(1) = -2 \).

This first term is important as it anchors the sequence, from which all subsequent terms are derived by consistently applying the common difference. Understanding the first term not only helps set up the structure of the sequence but also plays a critical role in calculating its overall sum.

Nth Term Formula

The Nth term formula is crucial for navigating arithmetic sequences, allowing one to find any specific term in the sequence without listing all previous terms. It’s an efficient tool provided by the formula:

• \( a_n = a_1 + (n-1) \, d \)

Here, \( a_n \) represents the nth term, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

For the sequence in question, where \( a_n = 1 - 3n \), understanding that the formula itself simplifies directly to \( a_n = 1 - 3n \) means we start from a defined first term and progress by subtracting 3 for each subsequent term.

This formula serves as a guide, enabling quick computation of any term in the sequence, which is particularly useful in large series or when determining specific terms to aid in sum calculations.