Problem 5 - College Algebra with Modeling and Visualization

Question

Let \(A_{n}\) represent the number of \(U . S .\) AIDS deaths reported \(n\) years after 2000 . Write a series whose sum gives the cumulative number of AIDS deaths from 2005 to 2009 .

Step-by-Step Solution

Answer

The series is \(A_5 + A_6 + A_7 + A_8 + A_9\).

Step 1: Understanding the Problem

We need to find the cumulative number of AIDS deaths from 2005 to 2009. This requires adding the annual deaths for each of those years.

Step 2: Express Years in Terms of n

The years 2005 to 2009 correspond to the values of \(n\) as follows: 2005 (\(n = 5\)), 2006 (\(n = 6\)), 2007 (\(n = 7\)), 2008 (\(n = 8\)), and 2009 (\(n = 9\)). We need to calculate the sum of \(A_n\) for \(n = 5\) to \(n = 9\).

Step 3: Writing the Series

The series representing the cumulative number of AIDS deaths from 2005 to 2009 is \( A_{5} + A_{6} + A_{7} + A_{8} + A_{9} \).

Step 4: Summing the Series

Add each term in the series: \(A_5\), \(A_6\), \(A_7\), \(A_8\), and \(A_9\). This will give the total number of deaths over the specified period.

Key Concepts

Series

A series is essentially the sum of a sequence of numbers. In mathematics, a series typically represents the addition of elements, which often come from a specific sequence. When we discuss the cumulative sum in a series, like in our exercise dealing with AIDS deaths from 2005 to 2009, we are adding up values described by a sequence over a certain period.
When you construct a series, you turn a list of terms from a sequence into a single value that indicates their total. For example, if you have a sequence representing annual deaths like \( \{A_5, A_6, A_7, A_8, A_9\} \), then their corresponding series is written as \( A_5 + A_6 + A_7 + A_8 + A_9 \).

• This results in a single number, which is the total of all the values added together.
• It's important because it gives us an aggregate measure instead of just individual years' data.

Understanding and calculating a series enables you to find the total sum of consecutive figures from a given sequence, which is particularly useful in statistics and financial assessments, among many other fields.

Sequence

Before you can talk about series, you need to first understand sequences. A sequence in mathematics is a set of numbers arranged in a specific order. In the context of the exercise, the sequence we're looking at is the number of AIDS deaths each year from 2005 to 2009, which we can denote as \( A_5, A_6, A_7, A_8, \) and \( A_9 \).

• Sequences can be finite or infinite, depending on whether they have an end or not.
• Our sequence is finite because it only covers specific years.

In this case, the sequence is derived from the number of years after 2000, hence each year's death count can be indexed to those specific years. This kind of sequence makes it straightforward to add these together to form a series.
Sequences not only help form series but are also essential to identifying patterns, which might be crucial for predicting future trends or diagnosing issues.

Algebra

Algebra plays a crucial role in forming and manipulating both sequences and series. It provides the tools needed to define and work with expressions for sequences and thereby calculate their corresponding series.

• For instance, identifying each year's death count from 2005 to 2009 involves defining those counts algebraically with expressions like \( A_n \), where \( n \) represents the number of years since 2000.
• In calculating the cumulative sum, algebraic skills become pivotal to accurately combine these values from the sequence into a coherent series.

Algebra is essentially the language of mathematics that helps describe real-world phenomena in a more formal and calculated manner.
By mastering algebra, you gain the ability to handle complex problems by breaking them into manageable parts, just as we analyzed the AIDS death counts across several years and aggregated them into a comprehensive total.