Problem 56 - Algebra 2

Question

CHALLENGE Determine whether each statement is true or false. If true, explain. If false, provide a counterexample. Every sequence is either arithmetic or geometric.

Step-by-Step Solution

Answer

False. The Fibonacci sequence is neither arithmetic nor geometric.

Step 1: Understanding Arithmetic Sequences

An arithmetic sequence is one where each term after the first is obtained by adding a constant difference to the preceding term. For example, the sequence 2, 4, 6, 8 is arithmetic as the difference between consecutive terms is 2.

Step 2: Understanding Geometric Sequences

A geometric sequence is one where each term after the first is obtained by multiplying the preceding term by a constant factor. For example, the sequence 3, 6, 12, 24 is geometric because each term is the previous term multiplied by 2.

Step 3: Checking the Key Statement

The statement claims that every sequence must either be arithmetic or geometric. To verify or refute this, consider the possibility of sequences that do not fit either category.

Step 4: Providing a Counterexample

Consider the sequence: 1, 1, 2, 3, 5, 8,... This is the Fibonacci sequence, where each term is the sum of the two preceding terms. This sequence is neither arithmetic (as there is no constant difference between terms) nor geometric (as there is no constant ratio between terms). Thus, it serves as a counterexample.

Key Concepts

Arithmetic Sequences

Arithmetic sequences are all about adding the same number over and over again. Start with any number. Then, just keep adding a fixed amount. Imagine lining up at a bus stop and someone joins the line every minute. That's an arithmetic sequence!

• Constant Difference: This is the key to arithmetic sequences. If each number is 3 more than the last, the constant difference is 3.
• Examples: 1, 4, 7, 10 has a constant difference of 3. It's like moving forward step by step.
• General Formula: If the first term is 'a' and the difference is 'd', a term can be given as: \( a_n = a + (n-1) \times d \).

Arithmetic sequences are common in daily life, like those amazing patterns you notice, such as staircase steps or calendar dates.

Geometric Sequences

In a geometric sequence, each term is found by multiplying the prior term by a constant number. Imagine a bank account where the money doubles each year. That's a geometric sequence thriving on multiplication!

• Common Ratio: This is the multiplier for going from one term to the next. If every term is twice the previous term, the common ratio is 2.
• Examples: 5, 10, 20, 40 is a sequence with a common ratio of 2. That's growth by multiplication, not addition like in arithmetic sequences.
• General Formula: With the first term as 'a' and a common ratio 'r', terms are represented by: \( a_n = a \times r^{(n-1)} \).

Look around, and you will find geometric sequences in how populations grow or how investments compound.

Fibonacci Sequence

The Fibonacci sequence is both simple and fascinating. Instead of relying on constant addition or multiplication, each term is a sum of the two preceding ones. It's like a handmade artisanal sequence!

• Unique Rule: Start with 0 and 1, then each number is the sum of the two numbers before it.
• Examples: Begins with 0, 1, then evolves into 0, 1, 1, 2, 3, 5, 8, and so on.
• Nature's Favorite: This sequence appears in shells, flowers, and other natural phenomena, creating patterns like spirals. Amazing!

The Fibonacci sequence is distinct from arithmetic and geometric sequences. It provides a perfect example that not all sequences fit neatly into simple categories.