Problem 9 - Precalculus with Limits

Question

In Exercises 5 - 16, determine whether the sequence is geometric. If so, find the common ratio. \( 1, -\dfrac{1}{2}, \dfrac{1}{4}, -\dfrac{1}{8}, \cdots \)

Step-by-Step Solution

Answer

Yes, the given sequence is geometric and the common ratio is -1/2.

Step 1: Understanding the sequence

Firstly, analyze the given sequence: \( 1, -\dfrac{1}{2}, \dfrac{1}{4}, -\dfrac{1}{8}, \cdots \). The sequence is changing signs and decreasing absolute values in each progressive term.

Step 2: Finding the common ratio

Let's try to find the ratio between each consecutive pair of numbers. For geometric sequences, the ratio should remain constant. Divide the second term(-1/2) by the first term(1). The result is -1/2. Now, divide the third term(1/4) by the second term(-1/2). The result is again -1/2. Perform this operation for every consecutive pair of terms in the sequence. In this case, since the sequence seems to follow a pattern, and the third term divided by the second term yields the same result, it indicates that we have found the common ratio of -1/2. Therefore, the sequence is indeed geometric, and the ratio remains constant.

Step 3: Concluding the exercise

Make a final conclusion: The given sequence: \( 1, -\dfrac{1}{2}, \dfrac{1}{4}, -\dfrac{1}{8}, \cdots \) is a geometric sequence and the common ratio is -1/2.

Key Concepts

Sequence

A sequence is a list of numbers arranged in a specific order. In mathematics, sequences can follow various patterns.
In our case, the sequence is:

• 1
• \(-\dfrac{1}{2}\)
• \(\dfrac{1}{4}\)
• \(-\dfrac{1}{8}\)
• \(\ldots\)

This particular sequence changes signs and its terms have decreasing absolute values.
In the world of sequences, it's important to know whether a sequence follows a specific structure, like arithmetic, geometric, or neither. Here, we'll determine if it's geometric.

Common Ratio

The common ratio is a critical aspect of a geometric sequence. In a geometric sequence, each term is obtained by multiplying the preceding term by a constant called the common ratio.
To find the common ratio, you divide any term in the sequence by its preceding term.
Let's calculate the common ratio for our sequence:

• Divide the second term \(-\dfrac{1}{2}\) by the first term \(1\) which gives us \(-\dfrac{1}{2}\).
• Then check by dividing the third term \(\dfrac{1}{4}\) by the second term \(-\dfrac{1}{2}\); the result is also \(-\dfrac{1}{2}\).

In this particular sequence, the common ratio is \(-\dfrac{1}{2}\). This consistency confirms the pattern as geometric.

Geometric Progression

A geometric progression is a sequence where each term is found by multiplying the previous term by the common ratio.
In simple words, it's a series of numbers with a constant multiplier.
For our sequence, each number appears as a result of multiplying the previous number by \(-\dfrac{1}{2}\).
Let’s see how it works for the sequence:

• Start with 1, multiply by \(-\dfrac{1}{2}\) to get \(-\dfrac{1}{2}\).
• Then, multiply \(-\dfrac{1}{2}\) by \(-\dfrac{1}{2}\) again to get \(\dfrac{1}{4}\).
• Repeat this process to find \(-\dfrac{1}{8}\) by multiplying \(\dfrac{1}{4}\) with \(-\dfrac{1}{2}\).

Thus, this sequence is confirmed to be a geometric progression with the common ratio \(-\dfrac{1}{2}\).