Problem 81 - A Graphical Approach to Precalculus with Limits

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) \(a_1 \approx 1169.06\), \(r = 0.916\). (b) \(a_{10} \approx 538.47\), \(a_{20} \approx 227\) indicate savings percentages.

Step 1: Calculate \(a_1\)

To find the value of \(a_1\), substitute \(n = 1\) into the given formula. This gives: \[a_1 = 1276(0.916)^1 = 1276 \times 0.916\]Calculate this value to find \(a_1\).

Step 2: Simplify \(a_1\)

Perform the multiplication: \[a_1 = 1276 \times 0.916 = 1169.056\]Thus, \(a_1 \approx 1169.06\).

Step 3: Identify the Common Ratio \(r\)

The common ratio \(r\) in the geometric sequence is given directly in the formula by the base of the exponent, which is 0.916. Therefore, \(r = 0.916\).

Step 4: Calculate \(a_{10}\)

Substitute \(n = 10\) into the formula to find \(a_{10}\): \[a_{10} = 1276(0.916)^{10}\]Compute this expression for \(a_{10}\).

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

First, calculate \((0.916)^{10}\), which is approximately 0.422. Then,\[a_{10} = 1276 \times 0.422 = 538.472\]This simplifies to \(a_{10} \approx 538.47\).

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

Substitute \(n = 20\) into the formula to find \(a_{20}\): \[a_{20} = 1276(0.916)^{20}\]Compute this expression for \(a_{20}\).

Step 7: Simplify \(a_{20}\)

First, calculate \((0.916)^{20}\), which is approximately 0.178. Then,\[a_{20} = 1276 \times 0.178 = 227\]This simplifies to \(a_{20} \approx 227\).

Step 8: Interpret \(a_{10}\) and \(a_{20}\)

The values \(a_{10} \approx 538.47\) and \(a_{20} \approx 227\) represent percentages of annual salary a person should have saved for retirement after 10 and 20 years, respectively, based on their moderate investment strategy.

Key Concepts

Investment Strategy

An investment strategy is a comprehensive plan that guides an individual's financial decisions and goals. Think of it as a roadmap that helps someone choose how to allocate their savings and investments over time. An optimal investment strategy is personal and aligned with one's risk tolerance, timeline, and financial aspirations.

• Risk Tolerance: This is how comfortable one is with potential fluctuations in the value of their investments. Those with a higher risk tolerance might pursue strategies with potentially higher returns but also higher risks.
• Diversification: Spreading investments across various asset classes like stocks, bonds, and real estate to manage risk.
• Time Horizon: The length of time someone expects to hold an investment before using the funds. Longer horizons allow for more aggressive strategies.
• Goals: Investment decisions should aim to help achieve specific financial goals, such as buying a house or saving for retirement.

By understanding and implementing an effective investment strategy, individuals can make informed decisions that align with their future needs and goals.

Retirement Planning

Retirement planning involves determining retirement income goals and the actions and decisions necessary to achieve those goals. It’s essential for ensuring a comfortable lifestyle post-employment. The idea is to secure financial stability in your retirement years.

• Savings Accumulation: Ensuring enough savings that will support you through the retirement period. This involves contributions to retirement accounts like 401(k)s and IRAs.
• Investments: Investing in various financial instruments to grow your savings over time. This might include stocks, bonds, and mutual funds.
• Budgeting: Evaluating expected retirement expenses and income to determine necessary savings. This also involves planning for unforeseen costs like healthcare.

With meticulous retirement planning, one can ensure they have the means to enjoy their leisure time without financial stress.

Common Ratio

In the context of geometric sequences, the common ratio is a crucial concept. It is the factor by which each term of the sequence is multiplied to obtain the next term. In mathematical terms, if the first term of a geometric sequence is denoted as \(a_1\) and the common ratio as \(r\), then the nth term is evaluated using the formula: \[a_n = a_1 imes r^{n-1}\]
Here are some important points about common ratio:

• The common ratio remains constant throughout the sequence. In this exercise, it's given as 0.916.
• When \(0 < r < 1\), the sequence is decreasing.
• If \(r > 1\), the sequence is increasing exponentially.

Understanding the common ratio is vital, as it helps analyze the progression of values in scenarios like savings growth, investments, and even solving real-world problems as seen in this retirement planning exercise.

Exponential Decay

Exponential decay is used to describe a quantity that decreases at a rate proportional to its current value over time. This concept appears in various fields and is essential to understand in contexts like finance and science.

• Definition: It's the process where values decrease by a consistent percentage rate per time unit. In the given formula, \(1276(0.916)^n\), the factor \(0.916\) signifies a decay each year.
• Applications: Beyond finance, exponential decay is pivotal in physics for radioactive decay, chemistry for reaction rates, and even population studies.
• Mathematical Representation: An exponential decay can be addressed using the formula: \[N(t) = N_0 imes e^{-kt}\] where \(N_0\) is the initial quantity and \(k\) is the decay rate.

Grasping exponential decay allows individuals to predict and understand the diminishing values in various contexts, from money to scientific phenomena.