Problem 6 - Algebra and Trigonometry

Question

Multiple Choice The sequence \(a_{1}=5, a_{n}=3 a_{n-1}\) is an example of a(n) _______ sequence. (a) alternating (b) recursive (c) Fibonacci (d) summation

Step-by-Step Solution

Answer

Recursive

Step 1 - Identify Given Information

The given sequence is defined as follows: the first term is given by \(a_{1} = 5\) and each subsequent term is determined by multiplying the previous term by 3, as indicated by \(a_{n} = 3a_{n-1}\).

Step 2 - Examine Definitions of Sequence Types

To classify the sequence, review the definitions: (a) Alternating sequence: alternates between positive and negative terms. (b) Recursive sequence: each term is defined using the previous term(s). (c) Fibonacci sequence: starts with 0 and 1, and each term is the sum of the two preceding ones. (d) Summation sequence: typically involves summing terms.

Step 3 - Determine the Correct Classification

Analyze the given sequence:- It does not alternate between positive and negative terms, so it is not (a) alternating.- It fits the description of a recursive sequence since each term is defined based on the previous term, making it (b) recursive.- It does not start with 0 and 1 or follow the Fibonacci pattern, ruling out (c) Fibonacci.- It does not fit the pattern of summing terms, ruling out (d) summation.

Key Concepts

Sequence Types

Understanding the different types of sequences is crucial for correctly identifying them in problems. Let's quickly go over some common sequence types:

• Alternating Sequence: This type of sequence alternates between positive and negative terms. For example, the sequence (-1, 2, -3, 4, -5,...) alternates signs.
• Recursive Sequence: Each term in the sequence is defined using the previous term(s). An example is the sequence given in the exercise: ( a_1 = 5, a_n = 3a_{n-1}).
• Fibonacci Sequence: This sequence starts with 0 and 1, and each subsequent term is the sum of the two preceding ones. The classic decimal Fibonacci sequence is (0, 1, 1, 2, 3, 5, 8, ...).
• Summation Sequence: Typically involves adding terms together. Unlike the recursive definition, this does not necessarily rely solely on previous terms. An example is the arithmetic sequence where terms are added to each subsequent value.

Knowing these basic definitions will help you quickly pinpoint the sequence type in any math problem.

Algebra

Algebra is a broad field of mathematics that often involves finding unknown variables using known values. In the context of sequences, algebra helps us create formulas and solve for unknown terms.

For instance, if given an arithmetic sequence, you might use an algebraic formula to find the sum of the first Nterms. Similarly, for geometric sequences (like the one in this problem), recognizing the algebraic relationship between terms enables you to describe the sequence succinctly:

• Geometric Sequence: Each term is found by multiplying the previous term by a constant factor (e.g., an = r * an-1). In this exercise, we repeatedly multiply by 3.

Algebraic manipulation is a powerful tool for understanding sequences, transforming equations, and proving properties of sequences.

Sequence Definitions

Defining sequences properly is essential when solving problems. Here's a quick guide to get you acquainted:

• First Term: The starting point of the sequence. In the problem, a_1 = 5 clearly states the first term.
• General Term: Expressed as a function of n, such as an = 3an-1» in this problem, it indicates how each subsequent term relates to the previous one.
• Pattern or Rule: The underlying rule that defines how the sequence progresses. Knowing this is key, whether it's adding, multiplying, or using more complex operations.

Ensuring you understand these definitions will make tackling sequence-related problems less daunting.

Recursive Formulas

Recursive formulas are critical in sequences where each term is defined relative to previous terms. Let’s break down what's been provided in the exercise:

• Initial Term: n={a_1 = 5}, l.
• Recursive Formula: a_n=3a_{n-1} denotes how each term is generated from the one before it.

The power of recursive sequences lies in their ability to describe even complex patterns with minimal information. Recursive sequences are widely used in computer science, especially in algorithms, to break down problems into simpler, more manageable parts. Remember: A well-stated initial term and a clear recursive formula are the keys to solving these types of problems.