Problem 17 - 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. $$3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \dots$$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio of \(\frac{1}{2}\).

Step 1: Identify Terms

The first step is to identify and write down the terms given in the sequence: \(a_1 = 3\), \(a_2 = \frac{3}{2}\), \(a_3 = \frac{3}{4}\), \(a_4 = \frac{3}{8}\).

Step 2: Check for Common Ratio

A sequence is geometric if there is a common ratio \(r\) between consecutive terms. Calculate the ratio of the second term to the first term: \(r = \frac{a_2}{a_1} = \frac{\frac{3}{2}}{3} = \frac{1}{2}\).

Step 3: Verify the Common Ratio

Check if the same ratio applies to the other pairs of consecutive terms.\[\frac{a_3}{a_2} = \frac{\frac{3}{4}}{\frac{3}{2}} = \frac{1}{2}\]

Step 4: Continue Verification for All Terms

Finally, check \(\frac{a_4}{a_3}\) to ensure consistency with the common ratio.\[\frac{a_4}{a_3} = \frac{\frac{3}{8}}{\frac{3}{4}} = \frac{1}{2}\] Since all ratios are equal, the sequence is geometric.

Step 5: Conclusion

Since the ratio between consecutive terms remains consistent at \(\frac{1}{2}\), the given sequence is a geometric sequence with a common ratio \(r = \frac{1}{2}\).

Key Concepts

Understanding the Common Ratio

The "common ratio" in a geometric sequence is key to determining whether a sequence is geometric. It is the factor you multiply by to move from one term of a sequence to the next. If this ratio remains constant between each pair of consecutive terms, then the sequence qualifies as geometric. For example, in our exercise, the sequence is:

• First term: \( 3 \)
• Second term: \( \frac{3}{2} \)
• Third term: \( \frac{3}{4} \)
• Fourth term: \( \frac{3}{8} \)

By calculating the ratio between each pair of terms, \( \frac{3/2}{3} = \frac{1}{2} \), we can see the ratio is consistently \( \frac{1}{2} \). Ensuring this consistency is what makes the sequence geometric.

Analyzing Consecutive Terms

"Consecutive terms" are terms that follow one after the other directly in a sequence. Understanding their relationship is essential when determining if a sequence is geometric. For instance, look at the terms \( 3 \) and \( \frac{3}{2} \). The step from the first to the second term by dividing by the common ratio \( \frac{1}{2} \) should match the step from the second to the third and so on.

• From \(3\) to \(\frac{3}{2}\): Divide by \(2\).
• From \(\frac{3}{2}\) to \(\frac{3}{4}\): Divide by \(2\).
• From \(\frac{3}{4}\) to \(\frac{3}{8}\): Divide by \(2\).

Noticing the consistent division by the common ratio across consecutive terms confirms the sequence follows the geometric sequence pattern.

Term Identification in Sequences

"Term identification" is the process of recognizing and writing down each term of a sequence. In any sequence-based problem, it is vital to identify the terms to understand their relationships. In the example given:

• First term: \( a_1 = 3 \)
• Second term: \( a_2 = \frac{3}{2} \)
• Third term: \( a_3 = \frac{3}{4} \)
• Fourth term: \( a_4 = \frac{3}{8} \)

This identification lays the groundwork for further analysis, such as checking for a common ratio. Proper term identification helps in applying mathematical operations uniformly, which is crucial when determining sequence properties like being geometric.