Problem 73 - A Graphical Approach to College Algebra

Question

Solve each problem . (Modeling) Investment for Retirement According to T. Rowe Price Associates, a person who has a moderate investment strategy and \(n\) years until retirement should have accumulated savings of \(a_{n}\) percent of his or her annual salary. The geometric sequence $$ a_{n}=1276(0.916)^{n} $$ gives the appropriate percent for each year \(n\) (a) Find \(a_{1}\) and \(r\) (b) Find and interpret the terms \(a_{10}\) and \(a_{20}\)

Step-by-Step Solution

Answer

(a) \(a_{1} = 1168.016\), \(r = 0.916\). (b) \(a_{10} = 554.744\) and \(a_{20} = 239.488\).

Step 1: Understanding the problem

This problem involves a geometric sequence that represents the percentage of an annual salary that should be saved based on years until retirement. We need to find specific terms of this sequence and interpret them.

Step 2: Identifying the sequence form

The given formula for the sequence is \(a_{n} = 1276(0.916)^{n}\). This indicates a geometric sequence with the initial term \(a_{0} = 1276\) and a common ratio of \(r = 0.916\).

Step 3: Calculating the first term \(a_{1}\)

We substitute \(n = 1\) into the formula to find \(a_{1}\): \ \(a_{1} = 1276(0.916)^1 = 1276 \times 0.916 = 1168.016\).

Step 4: Verifying the common ratio \(r\)

The common ratio, \(r\), given in the problem is 0.916. We check it by recognizing that each term multiplies the previous term by 0.916. Since no direct calculation was needed for this part, we accept \(r = 0.916\).

Step 5: Calculating \(a_{10}\)

To find \(a_{10}\), substitute \(n = 10\) into the sequence formula: \ \(a_{10} = 1276(0.916)^{10}\). Using a calculator, compute: \ \(a_{10} = 1276 \times 0.434 = 554.744\).

Step 6: Calculating \(a_{20}\)

To find \(a_{20}\), substitute \(n = 20\) into the formula: \ \(a_{20} = 1276(0.916)^{20}\). Using a calculator, compute: \ \(a_{20} = 1276 \times 0.188 = 239.488\).

Step 7: Interpretation of \(a_{10}\) and \(a_{20}\)

\(a_{10} = 554.744\) means that 10 years from retirement, a person with a moderate investment strategy should have saved about 554.744% of their annual salary. \(a_{20} = 239.488\) suggests that 20 years from retirement, the target savings should be 239.488% of the annual salary.

Key Concepts

Retirement Investment

Retirement investment is a strategy where individuals prepare financially for the years when they are no longer part of the workforce. It involves setting aside a portion of one's income over time to ensure a stable source of income during retirement years. Imagine you're on a long journey; retirement funds serve as the fuel needed to keep you moving. One popular method for retirement investment is using geometric sequences, as seen in our example. Here, the formula used is \(a_{n} = 1276(0.916)^n\), representing how much a person should save as a percentage of their annual salary. As the number \(n\) (years until retirement) decreases, the savings amount increases due to the geometric sequence properties. Key benefits of retirement investments include:

• Financial security in retirement years.
• Potential growth from compound interest and investments.

By understanding geometric sequences, one can more accurately predict and plan for the increasing savings needed as retirement approaches.

Savings Strategy

A savings strategy is a plan that outlines how and when to save money for future needs and plans. It's like setting milestones on a financial journey. In the context of geometric sequences, this strategy helps define how much one should save and when. The sequence \(a_{n} = 1276(0.916)^n\) serves as a blueprint to guide a person on how much of their salary to annually save based on the number of years until retirement. By understanding this sequence:

• You can compress years of savings data into a manageable formula.
• It helps visualize long-term financial goals.
• You translate abstract calculations into pragmatic savings actions.

Developing a savings strategy by understanding algebraic sequences provides clarity and control over one's financial future. It goes beyond saving a fixed percentage each year and dynamically adjusts based on the given number of years.

Algebraic Sequences

An algebraic sequence is a mathematical expression describing a sequence of numbers developing by a particular rule. For example, our problem is driven by a geometric sequence. Geometric sequences specifically involve numbers that multiply by a constant factor known as the common ratio \(r\). In our sequence \(a_{n} = 1276(0.916)^n\), the common ratio \(r = 0.916\) signifies the consistent percentage change each year. Here's why understanding this is beneficial:

• It provides an efficient method to calculate potential savings over time.
• It equips you with useful tools to understand financial models.
• It's highly applicable in various fields beyond finance, like physics and economics.

Recognizing algebraic sequences, particularly geometric ones, is crucial because it simplifies complex calculations into a series of manageable steps by consistently multiplying by the common ratio. This understanding can enhance one's ability to solve diverse mathematical and real-world problems efficiently.