Problem 53 - College Algebra

Question

How many terms must be added before the series \(-1-3-5-7 \ldots .\) has a sum less than \(-75 ?\)

Step-by-Step Solution

Answer

9 terms must be added for the sum to be less than -75.

Step 1: Identify the Sequence

The given series is an arithmetic series where each term decreases by 2. Therefore, the series can be expressed as \(-1, -3, -5, -7, \ldots\). The first term \(a = -1\) and the common difference \(d = -2\).

Step 2: Use the Formula for Sum of Arithmetic Series

The formula to find the sum of the first \(n\) terms of an arithmetic series is \( S_n = \frac{n}{2} (2a + (n-1)d) \). We want to find the smallest \(n\) such that \(S_n < -75\).

Step 3: Substitute Known Values into the Formula

Substitute \(a = -1\) and \(d = -2\) into the formula: \( S_n = \frac{n}{2} (2(-1) + (n-1)(-2)) \). Simplify this to \( S_n = \frac{n}{2} (-2 - 2n + 2) = \frac{n}{2} (-2n) = -n^2 \).

Step 4: Solve the Inequality

Now we have \(-n^2 < -75\). Solving for \(n\), we get \(n^2 > 75\). Taking the square root of both sides, \(n > \sqrt{75} \approx 8.66\). Since \(n\) must be an integer, the smallest possible \(n\) is 9.

Key Concepts

Sum of Arithmetic Series

In arithmetic series, the sum of the series involves adding a sequence of numbers where each number increases or decreases by a constant amount, known as the common difference. This consistent change simplifies our calculations and allows us to use a straightforward mathematical formula to find the sum of the series quickly.

To calculate the sum of the first \(n\) terms of an arithmetic series, we use the formula:

• \(S_n = \frac{n}{2} (2a + (n-1)d)\)

Here:

• \(S_n\) is the sum of the first \(n\) terms.
• \(a\) is the first term in the series.
• \(d\) is the common difference between the terms.

By substituting the known values into this formula, we can determine what the cumulative sum will be after a specified number of terms. This formula is invaluable when tackling problems where you need to find when the sum reaches certain values, especially when negativity or positivity matters, just as in the original problem.

Inequality Problem

An inequality problem, like the one given in the exercise, asks us to find out when a mathematical expression is less than, greater than, or equal to a certain value. In this instance, the goal is to find how many terms must be added to our sequence for the sum to become less than \(-75\).

To solve the inequality \(-n^2 < -75\), we first need to transform it into an equivalent inequality that's easier to handle:

• We multiply every term by \(-1\) to flip the inequality sign: \(n^2 > 75\).
• Then, we solve for \(n\) by taking the square root of both sides, giving us \(n > \sqrt{75} \approx 8.66\).

Since we're dealing with a sequence of terms, \(n\) must be a whole number. As a result, we round up to the nearest integer, which gives us \(n = 9\), ensuring our solution makes sense within the context of the sequence.

Series Convergence

Series convergence examines whether the terms of a series approach a specific value as more and more terms are added. However, for arithmetic series, convergence is a concept with a twist because arithmetic series have fixed differences and will continue to extend infinitely without converging to a finite value.

In this exercise, the notion of convergence is not about finding a limit the series approaches, but rather about understanding when certain conditions, such as a sum being less than a specified number, occur. Here we analyze when the cumulative sum of the series becomes meaningfully less, converging towards the condition set by the inequality. As we continue adding terms to the sequence \(-1, -3, -5, \ldots\), the sum does not approach a single limit, but rather continues to decrease indefinitely due to the negative common difference.

Sequence Analysis

Sequence analysis involves examining the properties or characteristics of a series of numbers and often forms the backbone of solving arithmetic series and other mathematical problems.

In our arithmetic sequence \(-1, -3, -5, -7, \ldots\), understanding the sequence is key. This well-defined structure arises from the first term and the common difference. The sequence strictly decreases, revealing patterns as it progresses.

Analyzing this sequence helps ensure we apply the correct formula and constraints, while noticing at which point certain conditions (like those dictated by inequalities) occur.

To properly analyze the sequence:

• Recognize the starting point ( \(-1\) in this case).
• Recognize the common difference ( \(-2\)).
• Apply understanding of how each subsequent term is derived.
• Relate terms to context-specific conditions, such as achieving sums or satisfying inequalities.

This type of analysis empowers us to predict how series behavior evolves over time, providing insights into when specific constraints are met.