Problem 93
Question
The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=-2 a_{n}, a_{0}=1 $$
Step-by-Step Solution
Verified Answer
The terms are: \( a_1 = -2 \), \( a_2 = 4 \), \( a_3 = -8 \), \( a_4 = 16 \), \( a_5 = -32 \).
1Step 1: Understand the Problem
We are given a recursively defined sequence \( \{ a_n \} \) with the initial condition \( a_0 = 1 \), and the recursive formula \( a_{n+1} = -2a_n \). We need to find the sequence terms \( a_1, a_2, a_3, a_4, \) and \( a_5 \).
2Step 2: Compute \( a_1 \)
Using the recursive formula, we calculate \( a_1 = -2a_0 = -2 imes 1 = -2 \).
3Step 3: Compute \( a_2 \)
Next, calculate \( a_2 = -2a_1 = -2 imes (-2) = 4 \).
4Step 4: Compute \( a_3 \)
Calculate \( a_3 = -2a_2 = -2 imes 4 = -8 \).
5Step 5: Compute \( a_4 \)
Find \( a_4 = -2a_3 = -2 imes (-8) = 16 \).
6Step 6: Compute \( a_5 \)
Finally, calculate \( a_5 = -2a_4 = -2 imes 16 = -32 \).
Key Concepts
Arithmetic ProgressionInitial ConditionsSequence Terms
Arithmetic Progression
Arithmetic progression, often abbreviated as AP, is a sequence of numbers in which the difference between consecutive terms, called the common difference, is constant. This means if you subtract one term from the next, the result is always the same. For example, in the sequence 2, 4, 6, 8, ..., the common difference is 2. This idea of a constant step makes arithmetic progressions a fundamental concept in series and sequences.
However, it is important to note that not all sequences are arithmetic progressions. In our original exercise, we are dealing with a recursive sequence rather than an arithmetic progression. Unlike arithmetic progression, where each term can be obtained through the addition of the previous term and a fixed number, recursive sequences use a formula involving previous terms to calculate subsequent terms. Understanding the difference between these types of sequences can help avoid confusion when solving related problems.
However, it is important to note that not all sequences are arithmetic progressions. In our original exercise, we are dealing with a recursive sequence rather than an arithmetic progression. Unlike arithmetic progression, where each term can be obtained through the addition of the previous term and a fixed number, recursive sequences use a formula involving previous terms to calculate subsequent terms. Understanding the difference between these types of sequences can help avoid confusion when solving related problems.
Initial Conditions
Initial conditions are parameters set at the beginning of a problem that define certain values of a sequence or system. In recursive sequences, these initial conditions provide the starting point. In the given problem, the initial condition is stated as \( a_0 = 1 \). It tells us the value of the sequence at the start, which helps us compute subsequent terms using the recursive formula provided.
Initial conditions are crucial because they anchor the sequence to a specific path. If they are changed, the whole sequence can be different. Simply put, without knowing where a sequence starts from, there's no real way to determine where it will go. So, in any problem involving sequences, always take note of any initial conditions provided before attempting to solve it.
Initial conditions are crucial because they anchor the sequence to a specific path. If they are changed, the whole sequence can be different. Simply put, without knowing where a sequence starts from, there's no real way to determine where it will go. So, in any problem involving sequences, always take note of any initial conditions provided before attempting to solve it.
Sequence Terms
Sequence terms are the individual elements or numbers that make up a sequence. In the context of the original exercise, these are the values of \( a_1, a_2, a_3, a_4, \) and \( a_5 \). Each sequence term is derived from the preceding term according to the key recursive formula \( a_{n+1} = -2a_n \).
Understanding how to calculate each term in a sequence is essential. As you may have seen, starting from the initial condition \( a_0 = 1 \), each term is computed by following the recursive relationship. This process of calculation is systematic:
Understanding how to calculate each term in a sequence is essential. As you may have seen, starting from the initial condition \( a_0 = 1 \), each term is computed by following the recursive relationship. This process of calculation is systematic:
- \( a_1 = -2 \times 1 = -2 \)
- \( a_2 = -2 \times (-2) = 4 \)
- \( a_3 = -2 \times 4 = -8 \)
- \( a_4 = -2 \times (-8) = 16 \)
- \( a_5 = -2 \times 16 = -32 \)
Other exercises in this chapter
Problem 91
In Problems 91-100, the sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=2 a_{n}, a_{0}=1 $$
View solution Problem 92
The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=2 a_{n}, a_{0}=3 $$
View solution Problem 94
The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=-2 a_{n}, a_{0}=2 $$
View solution Problem 95
The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=1+2 a_{n}, a_{0}=0 $$
View solution