Problem 15 - Precalculus Mathematics for Calculus

Question

Determine whether the sequence is geometric. If it 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 the terms of the sequence

The terms of the sequence are \(3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \dots\).

Step 2: Calculate the ratio between successive terms

To determine if a sequence is geometric, calculate the ratio between successive terms. The ratio \(r\) is given by \(r = \frac{a_{n+1}}{a_n}\).

Step 3: Calculate the ratio for the first two terms

For the terms \(3\) and \(\frac{3}{2}\), the ratio \(r = \frac{3/2}{3} = \frac{1}{2}\).

Step 4: Calculate the ratio for the second and third terms

For the terms \(\frac{3}{2}\) and \(\frac{3}{4}\), the ratio \(r = \frac{3/4}{3/2} = \frac{1}{2}\).

Step 5: Calculate the ratio for the third and fourth terms

For the terms \(\frac{3}{4}\) and \(\frac{3}{8}\), the ratio \(r = \frac{3/8}{3/4} = \frac{1}{2}\).

Step 6: Conclusion about the sequence

Since the ratio between successive terms is consistent at \(\frac{1}{2}\) across all comparisons, the sequence is geometric.

Key Concepts

Common Ratio

In the realm of geometric sequences, the common ratio is a crucial element. It's the factor that gives geometric sequences their unique nature. The common ratio is the number you multiply by to get from one term to the next. This constant multiplication factor is what sets a geometric sequence apart from other sequence types.
Here's how you determine it:

• Pick any two successive terms from the sequence, say the first term (\(a_n\)) and the second term (\(a_{n+1}\))
• Divide the second term by the first term: \(r = \frac{a_{n+1}}{a_n}\)

You will find that the result, called the common ratio (\(r\)), remains the same regardless of which pair of successive terms you choose. In our example sequence\(3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \dots\), the common ratio is consistently \(\frac{1}{2}\).
This consistency proves that the sequence is indeed geometric.

Successive Terms

Successive terms are simply the terms that come one after another in a sequence. Think of them as stepping stones in a number series, each one following the last according to the sequence's rule. In a geometric sequence, this rule is all about the common ratio.
Let's see how to work with successive terms in a geometric sequence:

• Start with your initial term, and keep multiplying by the common ratio to get the next.
• Continue this process to generate as many terms as needed.

In our sequence\(3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \dots\), each successive term is obtained by multiplying the preceding term by \(\frac{1}{2}\). This method of finding successive terms is consistent due to the presence of a common ratio.

Sequence Identification

Identifying whether a sequence is geometric involves checking for a consistent common ratio across all pairs of successive terms. This is the hallmark of geometric sequences and is essential for correct sequence identification.
Here's how to evaluate a sequence:

• Calculate the ratio of successive terms.
• If this calculated ratio is the same for all pairs, the sequence is geometric.

For our sequence\(3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \dots\), we checked several term pairs:\(\frac{3/2}{3} = \frac{1}{2}\),\(\frac{3/4}{3/2} = \frac{1}{2}\), and so on. Each time we found the ratio to be \(\frac{1}{2}\), confirming that it’s truly a geometric sequence. Remember, a consistent common ratio is what identifies a sequence as geometric.