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Chapter 11: Problem 65

If dollar 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.

Short Answer

Expert verified
(a) Terms: 1060, 1123.60, 1191.02, 1262.48, 1338.23. (b) The sequence is divergent.

Step by step solution

01

Calculate the First Term

To find the first term of the sequence, substitute \( n = 1 \) into the formula. \[ a_1 = 1000(1.06)^1 = 1000 \times 1.06 = 1060 \] Thus, the first term is \( 1060 \).
02

Calculate the Second Term

For the second term, substitute \( n = 2 \) into the formula. \[ a_2 = 1000(1.06)^2 = 1000 \times 1.1236 = 1123.60 \] The second term is \( 1123.60 \).
03

Calculate the Third Term

Substitute \( n = 3 \) into the formula to find the third term. \[ a_3 = 1000(1.06)^3 = 1000 \times 1.191016 = 1191.02 \] Thus, the third term is \( 1191.02 \).
04

Calculate the Fourth Term

Substitute \( n = 4 \) into the formula for the fourth term. \[ a_4 = 1000(1.06)^4 = 1000 \times 1.262477 = 1262.48 \] Hence, the fourth term is \( 1262.48 \).
05

Calculate the Fifth Term

To find the fifth term, plug \( n = 5 \) into the equation. \[ a_5 = 1000(1.06)^5 = 1000 \times 1.338225 = 1338.23 \] The fifth term is \( 1338.23 \).
06

Determine Convergence or Divergence

The sequence given by \( a_n = 1000(1.06)^n \) is an exponential growth sequence with a base greater than 1. This means that the terms of the sequence will grow indefinitely as \( n \) increases, causing the sequence to diverge.

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Key Concepts

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

Compounded Interest
Compounded interest is a method of calculating interest where the amount you earn is added back to the original principal, and interest is calculated on the new total in future periods. This creates a situation where you "earn interest on interest," which can significantly increase the total amount over time.
  • **Principal Amount (\(P\))**: This is your starting amount. In our example, it's $1000.
  • **Rate of Interest (\(r\))**: This is the interest rate per period. Here, it is 6% annually, which converts to 0.06 in decimal form.
  • **Number of Compounding Periods (\(n\))**: This refers to how often the interest is applied. In the context of our example, it compounds annually.
To calculate the compounded amount, the formula used is \( A_n = P (1 + r)^n \).
In our example, \( P = 1000 \), \( r = 0.06 \), and \( n \) is the number of years. This results in the sequence of amounts growing exponentially.
Exponential Growth
Exponential growth happens when the increase of a quantity is proportional to its current size, resulting in growth patterns that become steeper and steeper over time.
This is often expressed with base greater than 1 in its exponential form, such as \( (1.06)^n \) in our example.
  • **Exponential Function**: Given by \( a_n = A_0 imes b^n \), where \( b \) is the base of the exponential.
  • **Base Greater Than 1**: Ensures the quantity grows with \( n \). In our case \( 1.06 > 1 \) maintains the growth trend.
  • **Applications**: Used in finance for investments, population growth, and certain scientific scenarios, as it effectively models real-world growth situations.
In a situation like our exercise, the base \( 1.06 \) causes each term to increase over its predecessor, representing a typical exponential growth pattern. This results in larger increments as time goes by.
Convergent and Divergent Sequences
A sequence is a series of numbers defined in a specific order. Based on their behavior as the number of terms increases, sequences can be classified as convergent or divergent.
  • **Convergent Sequences**: The terms get closer and closer to a fixed value as \( n \) approaches infinity. They "converge" on this single value.
  • **Divergent Sequences**: The terms grow larger and do not settle at a fixed value. They "diverge" towards infinity or do not approach a specific limit.

The sequence in our example is \( 1000(1.06)^n \), which is divergent. This is because its base \( 1.06 \) is greater than 1, causing each term to grow indefinitely. Thus, as \( n \) increases, the sequence keeps expanding, demonstrating no sign of reaching a limit. Understanding this helps differentiate between situations that stabilize and those that continuously expand.

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Most popular questions from this chapter

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