Question

Table 15.2 shows data of AIDS deaths \(^{6}\) that occurred in the US where \(a_{n}\) is the number of deaths in year \(n\), and \(n=1\) corresponds to 2001 . (a) Find the partial sums \(S_{3}, S_{4}, S_{5}, S_{6}\). (b) Find \(S_{5}-S_{4}, S_{6}-S_{5}\) (c) Use your answer to part (b) to explain the value of \(S_{n+1}-S_{n}\) for any positive integer \(n .\) $$ \begin{array}{l|c|c|c|c|c|c} \hline \text { Year } & 2001 & 2002 & 2003 & 2004 & 2005 & 2006 \\ \hline n & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Deaths } & 17,402 & 16,948 & 16,690 & 16,395 & 16,268 & 14,016 \\\ \hline \end{array} $$

Short answer

Answer: The difference \(S_{n+1}-S_{n}\) represents the number of AIDS deaths that occurred in year n+1.

Step-by-step solution

Calculating Partial Sums

A partial sum is the sum of the first n terms of a sequence. In this case, we have the sequence of AIDS deaths per year from 2001 to 2006. The partial sums will be calculated as follows: \(S_{3}=a_{1}+a_{2}+a_{3}=17,402+16,948+16,690\) \(S_{4}=a_{1}+a_{2}+a_{3}+a_{4}=17,402+16,948+16,690+16,395\) \(S_{5}=a_{1}+a_{2}+a_{3}+a_{4}+a_{5}=17,402+16,948+16,690+16,395+16,268\) \(S_{6}=a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}=17,402+16,948+16,690+16,395+16,268+14,016\) Now, we compute the numerical values of these partial sums.

Computing the Numerical Values of Partial Sums

Calculating the sums, we get: \(S_{3}=17,402+16,948+16,690=50,040\) \(S_{4}=17,402+16,948+16,690+16,395=66,435\) \(S_{5}=17,402+16,948+16,690+16,395+16,268=82,703\) \(S_{6}=17,402+16,948+16,690+16,395+16,268+14,016=96,719\) Next, we will compute the differences between the consecutive partial sums as requested.

Computing Differences Between Consecutive Partial Sums

To compute the differences, we'll subtract consecutive partial sums: \(S_{5}-S_{4}=82,703-66,435\) \(S_{6}-S_{5}=96,719-82,703\) Now, we compute the numerical values of these differences.

Computing Numerical Values of Differences

Calculating the differences, we get: \(S_{5}-S_{4}=82,703-66,435=16,268\) \(S_{6}-S_{5}=96,719-82,703=14,016\) Lastly, let's interpret the meaning of the value of \(S_{n+1}-S_{n}\) for any positive integer \(n\).

Interpreting the Meaning of \(S_{n+1}-S_{n}\)

For any positive integer n, the difference \(S_{n+1}-S_{n}\) represents the change in the cumulative number of deaths between two consecutive years in the sequence. In other words, it tells us the number of AIDS deaths that occurred in year n+1. Looking at our computed values, we can see that \(S_{5}-S_{4}=16,268\), which is the number of AIDS deaths in 2005 (year 5), and \(S_{6}-S_{5}=14,016\), which is the number of AIDS deaths in 2006 (year 6). Thus, we can confirm that the difference \(S_{n+1}-S_{n}\) indeed represents the number of AIDS deaths that occurred in year n+1.

Key concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This fixed difference is known as the "common difference".
In the context of the provided exercise, the yearly recorded deaths can be viewed as terms in an arithmetic sequence if each year saw a consistent change in the number of deaths.
However, the data in the exercise does not form a perfect arithmetic sequence because the change in the number of deaths varies from year to year.

• The differences from 2001 through 2006 are not consistent, indicating that while the data may exhibit a trend, it does not strictly adhere to the pattern of an arithmetic sequence.
• In sequences where numbers grow or shrink at a steady rate, arithmetic sequence formulas can be handy. Yet, real-world data, such as yearly AIDS deaths, often demonstrate variability that deviates from such simplicity.

Summation

Summation is the process of adding a sequence of numbers to get their total. This process is particularly useful when dealing with partial sums, which are the sum of the first few terms of a sequence.
The partial sum can be mathematically expressed as:\[ S_n = a_1 + a_2 + ... + a_n \]where each \(a_i\) is a term in the sequence.
In the problem, partial sums \(S_3\), \(S_4\), \(S_5\), and \(S_6\) represent the accumulated number of AIDS deaths from year 2001 to respective years.

• For instance, \(S_3\) is the sum of deaths from 2001 to 2003: \(17,402 + 16,948 + 16,690 = 50,040\).
• The calculation of differences such as \(S_5 - S_4\) helps determine the number of deaths that occurred specifically in individual years.

Such summations and differences are essential in analyzing trends and changes over time.

Yearly Data Analysis

Yearly data analysis involves examining and interpreting data collected over a span of years. By doing so, one can discern patterns, trends, and insights that may not be immediately apparent from yearly figures alone.
In the provided exercise, the data of AIDS deaths over several years offers a glimpse into the changing landscape of the epidemic in the U.S.

• Comparing consecutive years, we notice a decline in AIDS-related deaths over time, with differences computed in the step solutions offering precise change values for each year.
• The specific analysis of \(S_{n+1}-S_n\), like \(S_5 - S_4\) and \(S_6 - S_5\), gives exact numbers, providing clarity on the effectiveness of health interventions or other external factors influencing the downward trend.

Such analysis is vital in public health and policy decision-making, as it helps identify where efforts should be concentrated to continue reducing the impact of diseases.