Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-14,\) determine whether each given sequence is geometric. If it is geometric, find \(r\) . If it is not geometric, explain why it is not. $$ 4,8,16,32,64, \dots $$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio \( r = 2 \).

Step 1: Define a Geometric Sequence

A sequence is geometric if each term after the first is the product of the previous term and a fixed number called the common ratio \( r \). This means for any terms \( a_n \) and \( a_{n+1} \) in the sequence, the ratio \( \frac{a_{n+1}}{a_n} \) should be constant.

Step 2: Identify Consecutive Terms

Identify and write down consecutive terms from the sequence: \( a_1 = 4, a_2 = 8, a_3 = 16, a_4 = 32, a_5 = 64 \).

Step 3: Calculate the Common Ratio

To find the common ratio \( r \), divide each term by the preceding term:- \( \frac{a_2}{a_1} = \frac{8}{4} = 2 \)- \( \frac{a_3}{a_2} = \frac{16}{8} = 2 \)- \( \frac{a_4}{a_3} = \frac{32}{16} = 2 \)- \( \frac{a_5}{a_4} = \frac{64}{32} = 2 \)

Step 4: Determine Consistency of the Ratio

All calculated ratios are equal, \( r = 2 \), indicating that the sequence has a constant common ratio.

Step 5: Conclusion of Geometric Sequence Verification

Since the ratio \( r \) is constant for all consecutive terms, the sequence \( 4, 8, 16, 32, 64, \dots \) is indeed geometric with a common ratio \( r = 2 \).

Key Concepts

Common Ratio

The common ratio is a fundamental concept in understanding geometric sequences. It serves as the constant multiplier that you apply to each term to get the next term in the sequence. The notion of a common ratio is crucial because it characterizes the nature of the sequence, differentiating geometric sequences from other types.

• The common ratio is denoted by the symbol \( r \).
• To find \( r \), divide any term in the sequence by its preceding term, such as calculating \( \frac{a_2}{a_1} \).
• In a geometric sequence, this division will yield the same value for every consecutive pair of terms.

If the common ratio is consistent across the sequence, then you have confirmed that it is geometric. For the given exercise, the ratio is consistently 2, indicating that each term is twice the previous one.

Consecutive Terms

Understanding the role of consecutive terms in a geometric sequence is key to determining whether the sequence is truly geometric. Consecutive terms are simply pairs of numbers that appear one after the other in the sequence.

• The concept revolves around comparing these terms to identify the sequence's behavior.
• To verify if a sequence is geometric, you must write down each consecutive term and calculate the ratio between them.
• For example, in the sequence 4, 8, 16, 32, 64, we looked at pairs like 8 and 4, 16 and 8, etc.

By ensuring that these consecutive terms have a consistent ratio, you verify the geometric nature of the sequence. This technique affirms whether the sequence grows at a constant multiplicative rate.

Constant Ratio

The key characteristic of a geometric sequence is the constant ratio. This is the factor that defines how each term relates to the previous term and remains unchanging throughout the entire sequence.

• Calculating a constant ratio involves dividing each term by the term before it.
• In a truly geometric sequence, this ratio should remain identical for every consecutive pair of terms.
• A sequence with a constant ratio is predictable, allowing you to generate future terms easily.

For example, in our sequence, both the first ratio \( \frac{8}{4} \) and all succeeding ratios like \( \frac{16}{8} \) equal 2. This constant multiplication factor confirms the sequence's geometric property, making calculations straightforward and future predictions straightforward.