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Chapter 13: Problem 42

A sequence is defined recursively. List the first five terms. \(a_{1}=-2 ; \quad a_{n}=n+3 a_{n-1}\)

Short Answer

Expert verified
The first five terms are: -2, -4, -9, -23, -64.

Step by step solution

01

- Identify the first term

The first term of the sequence is given directly: \[ a_{1} = -2 \]
02

- Find the second term

Use the recursive formula to find the second term: \[ a_{2} = 2 + 3a_{1} \] Substitute the value of \( a_{1} \): \[ a_{2} = 2 + 3(-2) = 2 - 6 = -4 \]
03

- Calculate the third term

Use the recursive formula to find the third term: \[ a_{3} = 3 + 3a_{2} \] Substitute the value of \( a_{2} \): \[ a_{3} = 3 + 3(-4) = 3 - 12 = -9 \]
04

- Determine the fourth term

Use the recursive formula to find the fourth term: \[ a_{4} = 4 + 3a_{3} \] Substitute the value of \( a_{3} \): \[ a_{4} = 4 + 3(-9) = 4 - 27 = -23 \]
05

- Establish the fifth term

Use the recursive formula to find the fifth term: \[ a_{5} = 5 + 3a_{4} \] Substitute the value of \( a_{4} \): \[ a_{5} = 5 + 3(-23) = 5 - 69 = -64 \]

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Key Concepts

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

Sequence Definition
A sequence is a list of numbers arranged in a specific order. Each number in the sequence is called a term. Sequences can be defined in different ways, such as explicitly by a formula or recursively. An explicit definition gives a direct formula for any term based on its position, while a recursive definition specifies each term based on previous terms. Recursive sequences are especially useful in modeling real-world problems where each term depends on prior values.
Recursive Formula
A recursive formula provides a way to find each term of a sequence by relating it to previous terms. For a sequence \(a_n\), the recursive formula typically has two parts:
  • Initial condition: one or more starting values.
  • Recurrence relation: a rule that defines \(a_n\) based on earlier terms in the sequence.
In our exercise, the sequence begins with \(a_1 = -2\), and each subsequent term \(a_n\) is calculated using the relation \ a_n = n + 3a_{n-1} \.
Term Calculation
Calculating the terms of a recursive sequence involves applying the recursive formula step by step. Let's break down the steps:
  • **First Term:** It's given directly as \(a_1 = -2\).
  • **Second Term:** Use the formula \(a_2 = 2 + 3a_1\). Substituting \(a_1 = -2\), we get \ a_2 = 2 + 3(-2) = -4 \.
  • **Third Term:** \(a_3 = 3 + 3a_2\). Using \ a_2 = -4 \, we find \ a_3 = 3 + 3(-4) = -9 \.
  • **Fourth Term:** \(a_4 = 4 + 3a_3\). Given \ a_3 = -9 \, \ a_4 = 4 + 3(-9) = -23 \.
  • **Fifth Term:** \ a_5 = 5 + 3a_4 \. Substituting \ a_4 = -23 \, we get \ a_5 = 5 + 3(-23) = -64 \.
Algebra
Algebra plays a crucial role in working with recursive sequences. By understanding algebraic operations, you can simplify and solve the recurrence relations. Here are some important points:
  • **Substitution:** This is used to replace a term with its known value to find the next term.
  • **Simplification:** Combining like terms and performing arithmetic operations correctly ensures accurate solutions.
  • **Identifying Patterns:** Recognize patterns in sequences which can sometimes help in finding shortcuts or validating calculations.
In our exercise, we repeatedly used substitution and simplification to find each subsequent term in the sequence. Mastery in basic algebra makes handling recursive sequences far easier.

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Most popular questions from this chapter

One method of pricing a stock is to discount the stream of future dividends of the stock. Suppose that a stock pays \(\$ P\) per year in dividends, and historically, the dividend has been increased \(i \%\) per year. If you desire an annual rate of return of \(r \%,\) this method of pricing a stock states that the price that you should pay is the present value of an infinite stream of payments: $$\text { Price }=P+P \cdot \frac{1+i}{1+r}+P \cdot\left(\frac{1+i}{1+r}\right)^{2}+P\cdot\left(\frac{1+i}{1+r}\right)^{3}+\cdots$$ The price of the stock is the sum of an infinite geometric series. Suppose that a stock pays an annual dividend of \(\$ 4.00\), and historically, the dividend has been increased \(3 \%\) per year. You desire an annual rate of return of \(9 \%\). What is the most you should pay for the stock?

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