Problem 15 - Precalculus: Mathematics for Calculus

Question

The first four terms of a sequence are given. Determine whether these terms can be the terms of a geometric sequence. If the sequence is geometric, find the common ratio. $$3072,1536,768,384, \dots$$

Step-by-Step Solution

Answer

Yes, it's geometric with a common ratio of \( \frac{1}{2} \).

Step 1: Understanding a Geometric Sequence

A geometric sequence has each term after the first as the product of the previous term and the common ratio. Mathematically, if a sequence \(a_1, a_2, a_3, \dots\) is geometric, then for some common ratio \(r\), we have \(a_{n} = a_{n-1} \times r\) for all \(n \).

Step 2: Comparing Consecutive Terms

Given sequence: 3072, 1536, 768, 384. Calculate the ratio of the second term to the first term: \( \frac{1536}{3072} = \frac{1}{2} \). Next, calculate the ratio of the third term to the second term: \( \frac{768}{1536} = \frac{1}{2} \). Finally, calculate the ratio of the fourth term to the third term: \( \frac{384}{768} = \frac{1}{2} \).

Step 3: Verifying Consistency of the Common Ratio

Since all the calculated ratios \( \frac{1}{2} \) are equal, it confirms that the sequence is geometric.

Step 4: State the Common Ratio

The consistent ratio between consecutive terms \( \frac{1}{2} \) is the common ratio of the geometric sequence.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key feature that distinguishes it. Each term in the sequence is multiplied by this common ratio to achieve the next term. For example, if you have the sequence terms 3072, 1536, 768, and 384, you can determine whether it is a geometric sequence by checking this ratio.
The common ratio can be found by dividing any term by its preceding term, such as:

• \( \frac{1536}{3072} = \frac{1}{2} \)
• \( \frac{768}{1536} = \frac{1}{2} \)
• \( \frac{384}{768} = \frac{1}{2} \)

Here, you see that the ratio \( \frac{1}{2} \) is consistent between terms. Thus, the common ratio of this geometric sequence is \( \frac{1}{2} \).

Identifying and verifying the common ratio helps confirm the sequence is indeed geometric.

Sequence Terms

Sequence terms are the individual numbers that make up a sequence. In a geometric sequence, these terms follow a specific pattern. You start with the first term and find subsequent terms by multiplying by a constant factor, known as the common ratio, which we discussed earlier.
For example, in the sequence 3072, 1536, 768, 384, each term is derived by multiplying the previous term by \( \frac{1}{2} \).

• The second term, 1536, is \( 3072 \times \frac{1}{2} = 1536 \).
• The third term, 768, is \( 1536 \times \frac{1}{2} = 768 \).
• The fourth term, 384, is \( 768 \times \frac{1}{2} = 384 \).

Recognizing this pattern allows you to predict upcoming sequence terms once the first term and common ratio are known.
This pattern keeps a sequence coherent and predictable.

Mathematical Sequence

A mathematical sequence is an ordered list of numbers that follow a particular rule. In the realm of mathematics, sequences can vary widely in type and complexity, with geometric sequences being just one of these categories.
A geometric sequence has a clear, defined rule for its progression. Each term is formed by multiplying the previous term by a constant called the common ratio. This particular type of sequence is useful for modeling exponential growth and decay, such as in finance or population studies.

• Starts with a first term, known as \( a_1 \).
• Proceeds by multiplying each term by the common ratio \( r \).

The sequence \( a_1, a_2, a_3, \ldots \) is defined by \( a_n = a_{n-1} \times r \) for all \( n \). For the sequence 3072, 1536, 768, 384, the common ratio is \( \frac{1}{2} \), confirming it is geometric.
Grasping the basic arrangement of these sequences helps in solving more complex, real-world problems.