Problem 56 - Contemporary Precalculus

Question

The sequence in which \(b_{n}=\) remaining life expectancy of a man at age \(n\) is approximately arithmetic. (a) Use the fact that a man's remaining life expectancy is 60.6 years at age 15 and 51.2 years at age 25 to find a formula for \(b_{n}\) (b) Determine the remaining life expectancy of a man at these ages: 20,22,30 and 40

Step-by-Step Solution

Answer

Answer: The remaining life expectancies for the given ages are: - Age 20: 55.9 years - Age 22: 54.02 years - Age 30: 46.5 years - Age 40: 37.1 years

Step 1: Determine the common difference

First, we need to find the common difference (d) between consecutive terms in the arithmetic sequence. Since \(b_{25}=51.2\) and \(b_{15}=60.6\), we have that: \[d = \frac{b_{25}-b_{15}}{25-15} = \frac{51.2-60.6}{10} = -\frac{9.4}{10} = -0.94\] The common difference is -0.94.

Step 2: Find the formula for \(b_n\)

With the common difference found, we can create the formula for the arithmetic sequence \(b_n\). Since \(b_{15}=60.6\), the formula for \(b_n\) can be written as follows: \[b_n = b_{15} + d(n-15)\] Plugging in our value for d, we get: \[b_n = 60.6 - 0.94(n-15)\]

Step 3: Calculate remaining life expectancy for specified ages

Now that we have the formula for \(b_n\) we can use it to find the remaining life expectancy for the requested ages: At age 20: \[b_{20} = 60.6 - 0.94(20-15) = 60.6 - 0.94(5) = 60.6 - 4.7 = 55.9\] At age 22: \[b_{22} = 60.6 - 0.94(22-15) = 60.6 - 0.94(7) = 60.6 - 6.58 = 54.02\] At age 30: \[b_{30} = 60.6 - 0.94(30-15) = 60.6 - 0.94(15) = 60.6 - 14.1 = 46.5\] At age 40: \[b_{40} = 60.6 - 0.94(40-15) = 60.6 - 0.94(25) = 60.6 - 23.5 = 37.1\] The remaining life expectancies for the given ages are: - Age 20: 55.9 years - Age 22: 54.02 years - Age 30: 46.5 years - Age 40: 37.1 years

Key Concepts

Remaining Life Expectancy

Remaining life expectancy is an estimation of the number of years a person is expected to live, starting from a certain age. In this exercise, we are exploring an arithmetic sequence where the remaining life expectancy for a man, denoted by \(b_n\), decreases as age increases. This real-life application of sequences helps us understand how life expectancies change with age. At age 15, the remaining life expectancy is 60.6 years, and at age 25, it is 51.2 years. By analyzing this information and treating it as an arithmetic sequence, we can calculate the expected remaining years of life for other ages. This gives us practical insights into aging and health planning.

Common Difference

In an arithmetic sequence, a common difference is a constant value by which consecutive terms differ. To find this common difference, \(d\), we subtract the value of a previous term from the subsequent term. For example, in our exercise, between ages 15 and 25, the common difference is calculated as:\[d = \frac{b_{25}-b_{15}}{25-15} = \frac{51.2-60.6}{10} = -0.94\]The negative sign indicates that life expectancy decreases as age increases. Understanding common difference is essential as it allows you to predict future terms in the sequence, which in this case, are more ages.

Sequence Formula

The sequence formula is a mathematical expression used to find any term in an arithmetic sequence. Once the common difference is known, you can create the sequence formula. Using the given information, we start with \(b_{15} = 60.6\). The formula can be written as:\[b_n = b_{15} + d(n-15)\]Plugging in the common difference \(d = -0.94\), we get:\[b_n = 60.6 - 0.94(n-15)\]This formula is powerful as it allows you to calculate the remaining life expectancy for any age \(n\) by just substituting the value of \(n\). It’s a practical tool for making predictions based on identified trends in data.

Age Specific Calculations

Using the sequence formula, we can easily calculate the remaining life expectancy for specific ages by substituting these ages into the formula. Let's see how it works with the examples given:

• At age 20: \[b_{20} = 60.6 - 0.94(20-15) = 55.9\] years
• At age 22: \[b_{22} = 60.6 - 0.94(22-15) = 54.02\] years
• At age 30: \[b_{30} = 60.6 - 0.94(30-15) = 46.5\] years
• At age 40: \[b_{40} = 60.6 - 0.94(40-15) = 37.1\] years

These results show how life expectancy decreases steadily as age increases. Understanding this calculation helps in planning for aspects like retirement and healthcare.