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Chapter 11: Problem 9

Find the common ratio. $$5, \frac{5 a}{2}, \frac{5 a^{2}}{4}, \frac{5 a^{3}}{8}, \dots$$

Short Answer

Expert verified
The common ratio is \(\frac{a}{2}\).

Step by step solution

01

Identify the Consecutive Terms

List the first two consecutive terms from the given sequence: \[5\] and \[\frac{5a}{2}\].
02

Set Up the Ratio

To find the common ratio \(r\), divide the second term by the first term: \[r = \frac{\frac{5a}{2}}{5}\].
03

Simplify the Expression

Simplify the fraction to find the common ratio: \[r = \frac{5a}{2} \div 5 = \frac{5a}{2} \times \frac{1}{5} = \frac{a}{2}\]. Therefore, the common ratio is \(\frac{a}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Ratio
In a geometric sequence, the common ratio is the factor by which we multiply one term to get the next term.
We represent this ratio as \( r \).
To find the common ratio between two consecutive terms, we divide the second term by the first term.
In the given sequence, let's take the first two terms: \( 5 \) and \( \frac{5a}{2} \).
We calculate the common ratio by setting up the fraction: \[ r = \frac{ \frac{5a}{2} }{ 5 } \].
After simplifying this fraction, as shown in the exercise solution, we get: \[ r = \frac{a}{2} \].
Hence, the common ratio here is \( \frac{a}{2} \). It's the key factor for determining the pattern of the sequence!
Sequence
A sequence is an ordered list of numbers, where each number is called a term.
In the case of a geometric sequence, we multiply each term by a common ratio to get the next one.
In our example, the sequence is: \( 5, \frac{5a}{2}, \frac{5a^2}{4}, \frac{5a^3}{8}, \dots \).
Each term after the first is obtained by multiplying the previous term by the common ratio \( \frac{a}{2} \).
This results in the sequence steadily changing based on the value of \( a \).
Geometric sequences can model many real-world scenarios like population growth, interest calculations, and much more!
Simplifying Fractions
Simplifying fractions means converting a complex fraction into its simplest form.
This process involves dividing the numerator and the denominator by their greatest common divisor (GCD).
In our exercise, during the calculation of the common ratio, we simplify: \[ r = \frac{5a}{2} \div 5 = \frac{5a}{2} \times \frac{1}{5} = \frac{a}{2} \].
Simplifying fractions helps to make calculations easier and the result more understandable.
Always check if both the numerator and the denominator have any common factors you can cancel out.
Simplification is an essential skill in math, helping to solve problems faster and more accurately.

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