Problem 9 - Algebra 2

Question

Is the sequence geometric? If so, find the common ratio and the next two terms. $$ 10,15,22.5,33.75, \dots $$

Step-by-Step Solution

Answer

Yes, the sequence is a geometric sequence. The common ratio is 1.5. The next two terms in the sequence are 50.625 and 75.9375.

Step 1: Test for Geometric Sequence

We can check if the sequence is geometric by dividing each term by the previous term. If the ratio is constant, then it is a geometric sequence. \[ \frac{15}{10} = 1.5, \frac{22.5}{15} = 1.5, \frac{33.75}{22.5} = 1.5 \] All these proportions are equal, indicating that this is indeed a geometric sequence.

Step 2: Identify Common Ratio

In a geometric sequence, the common ratio is the constant value we multiply to get from each term to the next. Here, the common ratio is 1.5, as shown in Step 1.

Step 3: Find the Next Two Terms

To find the next terms, we simply multiply the last known term by the common ratio. Therefore, the next term is \[ 33.75 \times 1.5 = 50.625 \] And the term after that is \[ 50.625 \times 1.5 = 75.9375 \] So, the next two terms of the sequence are 50.625 and 75.9375.

Key Concepts

Common Ratio

In the context of a geometric sequence, the common ratio plays a crucial role. It is the constant factor that connects each term to the next in the sequence. In simple terms, if you can multiply each term by the same number to get to the next term, you're dealing with a geometric sequence! The number you multiply with is known as the common ratio.

• Formula to find the common ratio: Divide any term by the previous term.
• Example in action: For the sequence 10, 15, 22.5, 33.75, calculating the common ratio is done as follows: \( \frac{15}{10} = 1.5 \).
• Consistency check: Ensure that every term divided by its predecessor yields the same number (in our case, 1.5).

If the ratio is the same for all pairs of consecutive terms, you've confirmed it is indeed a geometric sequence. Understanding the common ratio not only tells us about the relationship between terms but also enables us to generate further terms in the series with ease.

Terms of a Sequence

Understanding the terms within a geometric sequence means recognizing how each number is derived from the preceding one. Each term in a geometric sequence can be expressed as:

• \( a_n = a_1 \times r^{(n-1)} \)

where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the position of the term in the sequence. The predictability of the structure makes it easy to find any term you want, whether it is close or far from the starting point.
By knowing this formula, you could find not just the next one or two terms but any term in the sequence. For example, we already calculated the next two terms: 50.625 and 75.9375. Using the formula, predicting what the 10th or 20th term will be becomes straightforward. The methodical nature of sequences within mathematics demonstrates just one example of the overall order within math. Each term's reliance on multiplication by the common ratio keeps the sequence's pattern going forward.

Mathematics Problem Solving

Problem-solving in mathematics is an essential skill that can be deeply engaging. The process allows you to apply logical thinking to reach a solution.
For geometric sequences, problem-solving often involves:

• Recognizing the sequence type - Is it geometric, arithmetic, or neither?
• Finding the common ratio - Confirm that it's consistent to validate a geometric sequence.
• Applying formulas correctly - Use \( a_n = a_1 \times r^{(n-1)} \) to find any term in the sequence.
• Performing calculations accurately - Ensure that multiplication (as needed for the geometric sequences) is done precisely to avoid errors.

Through these steps, you enhance your mathematical reasoning and develop proficiency in handling different types of problems. The repetition builds confidence and understanding, solidifying your ability to solve not just the problem at hand but similar ones in the future. Mathematics isn't just about right and wrong answers—it's about the journey of understanding patterns and logic behind each problem.