Problem 16 - Algebra and Trigonometry

Question

Determine whether the sequence is geometric. If is geometric, find the common ratio. $$ 27,-9,3,-1, \ldots $$

Step-by-Step Solution

Answer

Yes, the sequence is geometric with \(-\frac{1}{3}\) as the common ratio.

Step 1: Identify Terms in Sequence

The given sequence is 27, -9, 3, -1, ... We will check if it is a geometric sequence.

Step 2: Check for a Common Ratio

A sequence is geometric if there is a constant ratio between consecutive terms. Calculate the ratio for the first two terms: \(-9 \div 27 = -\frac{1}{3}\).

Step 3: Verify the Common Ratio

Calculate the ratio for the second and third terms: \(3 \div -9 = -\frac{1}{3}\). This matches our previous ratio.

Step 4: Validate with Additional Terms

Calculate the ratio for the third and fourth terms: \(-1 \div 3 = -\frac{1}{3}\). This confirms the common ratio is constant.

Key Concepts

Common Ratio

In order to understand geometric sequences, we need to talk about the "common ratio." Imagine you are baking cookies, and you find a secret ingredient that everyone loves. You add the same amount of this ingredient each time you bake. The common ratio in a geometric sequence works similarly. It is the factor you multiply with one term to get the next one.

Mathematically, you find this ratio by dividing any term by the term before it. Using our sequence example: 27, -9, 3, -1,... when we divide

• the second term by the first: o(-9) ÷ 27 = -\(\frac{1}{3}\)
• the third term by the second: o3 ÷ -9 = -\(\frac{1}{3}\)
• the fourth term by the third: o-1 ÷ 3 = -\(\frac{1}{3}\)

In this sequence, the common ratio is constant: -\(\frac{1}{3}\). Understanding the common ratio helps you identify and work with geometric sequences effectively.

Geometric Sequence Identification

To determine if a sequence is geometric, we need to know that a geometric sequence is one where each term is obtained by multiplying the previous term by a fixed, non-changing number, known as the common ratio. Let's break this down further.

The sequence we're examining: 27, -9, 3, -1,... must fulfill this criterion of having a constant multiplier between every pair of consecutive terms.

• The first step is to clearly note down the terms of the sequence.
• Next, you test whether a consistent multiplier exists between them by calculating the ratio of consecutive terms.

For instance, if you find that multiple consecutive divisions give you -\(\frac{1}{3}\), then you have identified a geometric sequence. Identifying a geometric sequence is a stepping stone toward solving more complex problems. It often involves simple arithmetic yet reveals profound mathematical patterns.

Constant Ratio Verification

After determining the potential common ratio, consistency across the sequence must be verified. Constant ratio verification ensures that the sequence truly represents a geometric pattern.

To confirm, calculate the ratio between multiple pairs of consecutive terms:

1. For the first set, calculate the ratio: o(-9) ÷ 27 = -\(\frac{1}{3}\)
2. For the next set, check: o3 ÷ -9 = -\(\frac{1}{3}\)
3. And again for the next set:o-1 ÷ 3 = -\(\frac{1}{3}\)

If all ratios are identical, you have successfully verified the common ratio. This method guarantees that the sequence is indeed geometric. Striving for this verification reinforces the accuracy of your identification, solidifying your understanding of the mathematical concept.