Problem 33
Question
If \(\$ 1000\) is invested at 6\(\%\) interest, compounded annually, then after \(n\) years the investment is worth \(a_{n}=1000(1.06)^{n}\) dollars. (a) Find the first five terms of the sequence \(\left\\{a_{n}\right\\} .\) (b) Is the sequence convergent or divergent? Explain.
Step-by-Step Solution
Verified Answer
First five terms are 1060, 1123.6, 1191.016, 1262.477, 1338.226. Sequence is divergent.
1Step 1: Calculate First Term
To find the first term of the sequence, plug in \(n = 1\) into the formula. \[ a_1 = 1000(1.06)^1 = 1000 \times 1.06 = 1060 \]
2Step 2: Calculate Second Term
To find the second term of the sequence, plug in \(n = 2\) into the formula. \[ a_2 = 1000(1.06)^2 = 1000 \times 1.1236 = 1123.6 \]
3Step 3: Calculate Third Term
To find the third term of the sequence, plug in \(n = 3\) into the formula. \[ a_3 = 1000(1.06)^3 = 1000 \times 1.191016 = 1191.016 \]
4Step 4: Calculate Fourth Term
To find the fourth term of the sequence, plug in \(n = 4\) into the formula. \[ a_4 = 1000(1.06)^4 = 1000 \times 1.262477 = 1262.477 \]
5Step 5: Calculate Fifth Term
To find the fifth term of the sequence, plug in \(n = 5\) into the formula. \[ a_5 = 1000(1.06)^5 = 1000 \times 1.3382256 = 1338.2256 \]
6Step 6: Determine Sequence Convergence
To determine if the sequence is convergent, consider the growth rate. The formula \( a_n = 1000(1.06)^n \) represents exponential growth since the base \(1.06 > 1\). Therefore, the sequence is divergent as it will continue to increase indefinitely.
Key Concepts
Compound InterestSequence DivergenceInvestment GrowthCalculus
Compound Interest
Compound interest is a powerful concept in finance that helps grow investments faster than simple interest. It refers to earning interest on both the initial principal and any accrued interest from previous periods. This results in exponential growth over time.
In our example, an initial investment of $1000 earns a 6% annual interest. This means that each year, the investment's value multiplies by 1.06 (since 6% is represented as 0.06 in decimal form, and we add 1 to account for the original principal).
Using the formula \( a_n = 1000(1.06)^n \), we can calculate the investment's value for each year, where \( n \) represents the number of years. This formula illustrates how compound interest works to increase an investment's value exponentially over time.
In our example, an initial investment of $1000 earns a 6% annual interest. This means that each year, the investment's value multiplies by 1.06 (since 6% is represented as 0.06 in decimal form, and we add 1 to account for the original principal).
Using the formula \( a_n = 1000(1.06)^n \), we can calculate the investment's value for each year, where \( n \) represents the number of years. This formula illustrates how compound interest works to increase an investment's value exponentially over time.
Sequence Divergence
In mathematics, a sequence is a list of numbers arranged in a specific order based on a formula or rule. Whether a sequence is convergent or divergent depends on its behavior as it progresses toward infinity.
A sequence converges if its terms approach a finite limit as the sequence progresses. Conversely, it diverges if the terms continue to increase or decrease without bound.
In our problem, the sequence \( a_n = 1000(1.06)^n \) is identified as divergent. Since the base of the exponential term \( 1.06 \) is greater than 1, the sequence will continue to grow indefinitely. The terms do not stabilize toward a finite number but rather increase without limit.
A sequence converges if its terms approach a finite limit as the sequence progresses. Conversely, it diverges if the terms continue to increase or decrease without bound.
In our problem, the sequence \( a_n = 1000(1.06)^n \) is identified as divergent. Since the base of the exponential term \( 1.06 \) is greater than 1, the sequence will continue to grow indefinitely. The terms do not stabilize toward a finite number but rather increase without limit.
Investment Growth
Investment growth is the increase in value of an asset over time. In the context of our exercise, we consider how an investment grows with compound interest.
The formula \( a_n = 1000(1.06)^n \) offers a practical way to calculate future values of an investment as it appreciates each year. The exponential nature of this formula demonstrates how investments can grow significantly over extended periods of time.
This growth is not merely a fixed increase each year. Instead, each year's growth builds on the previous year's, amplifying over time. This is a hallmark of exponential growth, where the potential for substantial increases in value becomes evident with longer time horizons.
The formula \( a_n = 1000(1.06)^n \) offers a practical way to calculate future values of an investment as it appreciates each year. The exponential nature of this formula demonstrates how investments can grow significantly over extended periods of time.
This growth is not merely a fixed increase each year. Instead, each year's growth builds on the previous year's, amplifying over time. This is a hallmark of exponential growth, where the potential for substantial increases in value becomes evident with longer time horizons.
Calculus
Calculus is a branch of mathematics focusing on the change and motion. It includes concepts such as differentiation and integration, which help analyze and understand variable rates of change.
In relation to exponential growth and sequences, calculus can aid in understanding the continuous growth models. Although the investment example given is not continuous but grows annually, calculus provides the tools to comprehend and evaluate how functions behave and change over time.
This can be particularly useful for analyzing real-world problems like compound interest, where growth is not linear but exponential, and foreseeing the future value of investments requires understanding these changes effectively.
In relation to exponential growth and sequences, calculus can aid in understanding the continuous growth models. Although the investment example given is not continuous but grows annually, calculus provides the tools to comprehend and evaluate how functions behave and change over time.
This can be particularly useful for analyzing real-world problems like compound interest, where growth is not linear but exponential, and foreseeing the future value of investments requires understanding these changes effectively.
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