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Chapter 8: Problem 22

Find the first four terms of each sequence. $$a_{1}=-1, a_{n}=a_{n-1}-4, \text { for } n>1$$

Short Answer

Expert verified
The first four terms are -1, -5, -9, and -13.

Step by step solution

01

Understand the sequence formula

The sequence is given by the initial term \( a_1 = -1 \) and the recursive formula \( a_n = a_{n-1} - 4 \) for \( n > 1 \). This means each term is obtained by subtracting 4 from the previous term.
02

Find the first term of the sequence

The first term is given directly as \( a_1 = -1 \).
03

Calculate the second term

Using the recursive formula, calculate the second term: \( a_2 = a_1 - 4 = -1 - 4 = -5 \).
04

Calculate the third term

Use the recursive formula to find the third term: \( a_3 = a_2 - 4 = -5 - 4 = -9 \).
05

Calculate the fourth term

Again, apply the recursive formula to find the fourth term: \( a_4 = a_3 - 4 = -9 - 4 = -13 \).

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Key Concepts

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

Sequence Formula
In mathematics, sequences are ordered lists of numbers where the order and pattern of numbers follow a specific logic. The sequence formula is a way to describe how to find any term in a sequence.
  • There are different types of sequences, such as arithmetic, geometric, and more. Each has its own distinctive formulas.
  • In the given exercise, the sequence is described by the formula for the first term, \( a_1 = -1 \), and a recursive relationship for subsequent terms.
The formula \( a_n = a_{n-1} - 4 \) is used to find subsequent terms after the initial term. This means to find any term \( a_n \) in this sequence, you would subtract 4 from the previous term \( a_{n-1} \). This formula indicates an arithmetic sequence where each term decreases by 4 from the prior term.
Recursive Formula
A recursive formula specifies each term in a sequence using the previous term(s). This is a powerful method, particularly for sequences where each term relates closely to its predecessors.
  • The exercise provides us with a recursive formula: \( a_n = a_{n-1} - 4 \), applied for \( n > 1 \).
  • Recursive formulas require you to know the initial term to generate future terms. Here, \( a_1 = -1 \) serves as the starting point.
To practically use a recursive formula, after identifying \( a_1 \), proceed by using the formula to calculate \( a_2 \), then \( a_3 \), and continue similarly. This approach processes the sequence step-by-step, allowing you to determine each subsequent term as shown in this exercise.
Sequence Terms
Sequence terms are the individual numbers in a sequence, each labeled with an index. The index helps in identifying the position of the term in the sequence (e.g., \( a_1 \), \( a_2 \), \( a_3 \), etc.).
  • The first term or initial term is explicitly given in this exercise, \( a_1 = -1 \).
  • Using the recursive formula, further terms can be calculated one by one.
  • In this sequence, the next terms are derived by continually subtracting 4 from the previous term.
For example, the second term \( a_2 \) is \( -5 \), the third term \( a_3 \) is \( -9 \), and the fourth term \( a_4 \) is \( -13 \). In any sequence, each term builds upon the information or values provided by preceding terms, highlighting the connected nature of sequence terms.

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Most popular questions from this chapter

Use a formula to find the sum of each arithmetic series. $$3+5+7+9+11+13+15+17$$

Use a formula to find the sum of each arithmetic series. $$7.5+6+4.5+3+1.5+0+(-1.5)$$

Find \(a_{1}\) for each arithmetic sequence. $$a_{5}=27, a_{15}=87$$

Use a formula to find the sum of each arithmetic series. $$1+2+3+4+\dots+50$$

Solve each problem . (Modeling) Investment for Retirement According to T. Rowe Price Associates, a person who has a moderate investment strategy and \(n\) years until retirement should have accumulated savings of \(a_{n}\) percent of his or her annual salary. The geometric sequence $$ a_{n}=1276(0.916)^{n} $$ gives the appropriate percent for each year \(n\) (a) Find \(a_{1}\) and \(r\) (b) Find and interpret the terms \(a_{10}\) and \(a_{20}\)
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