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Chapter 7: Problem 5

In Exercises \(4-8,\) assume that a person invests \(\$ 2000\) at 14 percent interest compounded annually. Let \(A_{n}\) represent the amount at the end of \(n\) years. Find an initial condition for the sequence \(A_{0}, A_{1}, \ldots\)

Short Answer

Expert verified
\$2000

Step by step solution

01

Understand the Problem

Here the sequence \(A_{0}, A_{1}, \ldots\) represents the amount of money at the end of each year for \(n\) years, when an investment of $2000 is made at an interest rate of 14%, compounded annually.
02

Identify the Initial Condition

The initial condition of a sequence is usually the very first term of the sequence, which is when \(n = 0\). In this context, the initial condition would be the amount of money at the start, before any interest was computed.
03

Calculate the Initial Condition

Since the initial condition is the starting amount before any interest is applied, i.e., when \(n = 0\), it would be equal to the initial invest, which is \(A_0 = $2000\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Initial Condition
When dealing with sequences related to financial investments, the initial condition is a key starting point. In the context of compound interest, the initial condition represents the initial amount of money invested, before any interest has been added.
This is essentially the first term of the sequence, denoted typically as \(A_0\).

In the given exercise, the initial condition is the amount of money you start with, which is \(\$2000\).
  • Why it's important: The initial condition serves as the basis for calculating future amounts after interest accumulates.
  • Link to sequences: Knowing the initial condition helps you construct the entire sequence of an investment over time.
Once you understand the initial condition, it's easier to track how the investment grows annually with compounding interest.
Sequence of Investments
A sequence in mathematics often refers to a set of numbers arranged in a specific order. In financial contexts, like our investment example, sequences help us model how money grows over time.
Each term in the sequence \(A_0, A_1, A_2, \ldots\) corresponds to the amount of money at the end of each year.

Here's how you typically understand this:
  • Start Point: Begin with the initial condition \(A_0 = \$2000\).
  • Growth Calculation: Each subsequent term \(A_n\) is influenced by the compounding interest.
  • Incremental Terms: They follow the formula \(A_{n+1} = A_n \times (1 + r)\), where \(r\) is the interest rate.
This forms a simple, repeatable pattern鈥攖he essence of a sequence. In our case, each year, the investment grows by an additional 14% based on the previous year's amount.
Annual Compounding of Interest
Compounding interest annually means that the interest earned each year is added to the principal amount for the next year's interest calculation.
In our example, the interest rate is 14%, so each year, 14% of the current value is added back to the investment.

Here's how it impacts the investment:
  • Initial Growth: The first year, the investment grows from \(\\(2000\) to \(\\)2000 \times 1.14\) (14% increase).
  • Subsequent Growth: In following years, the new amount becomes the base for calculating the next year's growth.
  • Exponential Increase: Due to compounding, even small interest rates can significantly increase the total amount over time.
This process can result in exponential growth, illustrating why compound interest is such a powerful tool in finance.

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