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Chapter 8: Problem 43

Write an explicit rule for the sequence. $$ a_1=-2, a_n=3 a_{n-1} $$

Short Answer

Expert verified
The explicit rule for the sequence is \( a_n = -2*3^{(n-1)} \).

Step by step solution

01

Understand the Sequence

The sequence begins with -2. Each following term of the sequence is obtained by multiplying the previous term by 3. Hence, a few terms of the sequence are \( a_1 = -2 \), \( a_2 = -2*3 = -6 \), \( a_3 = -6*3 = -18 \), and so on.
02

Identify Pattern for General Form

Analyzing the sequence, it is noticed that \( a_2 = -2*3^1 \), \( a_3 = -2*3^2 \). Observing the pattern we can deduce that the \( n^{th} \) term of the sequence, \( a_n \), is given by \( -2*3^{(n-1)} \)
03

Write General Explicit Rule

The explicit rule for any \( n^{th} \) term of the sequence is therefore given by \( a_n = -2*3^{(n-1)} \).

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Key Concepts

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

Geometric Sequence
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the given problem, the sequence starts with \( a_1 = -2 \), and each subsequent term is obtained by multiplying the previous term by 3. This makes the common ratio \( r = 3 \).
  • Starting Term \( (a_1) \): This is the first term of the sequence, \( -2 \) in this case.
  • Common Ratio \( (r) \): The number by which we multiply each term to get the next, which is 3 here.
  • General Form: The general formula for any geometric sequence is \( a_n = a_1 \cdot r^{(n-1)} \).
From this, we derive the explicit formula: \( a_n = -2 \cdot 3^{(n-1)} \). This formula allows us to calculate any term in the sequence without having to know the preceding term.
Recurrence Relation
A recurrence relation in a sequence defines how each term relates to the previous term(s). For our geometric sequence, the recurrence relation is \( a_n = 3a_{n-1} \). This means each term is produced by taking the previous term, \( a_{n-1} \), and multiplying it by 3.
  • Previous Term: \( a_{n-1} \) refers to the term directly before \( a_n \).
  • Derived Term: To find \( a_n \), simply multiply \( a_{n-1} \) by the common ratio (3 in this case).
Recognizing recurrence relations is key to understanding how sequences evolve step by step. In applying the recurrence relation step by step, you can derive or verify each term of the sequence.
Algebraic Patterns
Algebraic patterns are mathematical expressions that represent a sequence or a data set in algebraic terms. These patterns help in understanding the sequence's behavior over time. In our example, the sequence \(-2, -6, -18, -54, \ldots\) shows a consistent pattern: each term is three times the preceding term.
  • Recognizing Patterns: Start with the known terms and observe how they relate to each other. Determine the constant operations, like multiplication by a common ratio.
  • Using Observations: Once the pattern is clear, it can be represented algebraically. In this problem, the pattern gave rise to the formula \( a_n = -2 \cdot 3^{(n-1)} \).
  • Utility: Algebraic patterns allow you to predict future values and understand the underlying structure of the sequence.
Analyzing these patterns is critical for deriving explicit formulas and understanding how sequences behave.
Sequence Formulas
Sequence formulas provide a practical means of calculating any term in a sequence without going through all the preceding terms. In this exercise, we focused on the explicit formula for a geometric sequence.
  • Explicit Formula: For a geometric sequence, this is given by \( a_n = a_1 \cdot r^{(n-1)} \). It directly computes the nth term based on its position \( n \).
  • Convenience and Efficiency: Instead of calculating each term serially using the recurrence relation, the explicit formula jumps directly to the desired term.
  • Applications: Useful in various scenarios including finance, computing powers of interest, and growth modeling, where you need quick access to specific elements of long sequences.
Mastering the creation and use of sequence formulas empowers you to solve complex problems in a straightforward manner.

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Most popular questions from this chapter

In Exercises 47鈥�52, find the sum. $$ \sum_{i=1}^9 6(7)^{i-1} $$

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MAKING AN ARGUMENT Your friend believes the sum of a series doubles when the common difference of an arithmetic series is doubled and the first term and number of terms in the series remain unchanged. Is your friend correct? Explain your reasoning.

write a rule for the nth term of the arithmetic sequence. \(a_5=41, a_{10}=96\)
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