Natalia is writing a recursive formula to represent the sequence.
[tex]$8,12,18,27, \ldots$[/tex]
What value should she use as the common ratio in the formula?
A. [tex]$\frac{1}{4}$[/tex]
B. [tex]$\frac{2}{3}$[/tex]
C. [tex]$\frac{3}{2}$[/tex]
D. [tex]$\frac{4}{1}$[/tex]
Answer (1)
Calculate the ratio between the second and first terms: 8 12 = 2 3 .
Calculate the ratio between the third and second terms: 12 18 = 2 3 .
Calculate the ratio between the fourth and third terms: 18 27 = 2 3 .
The common ratio is 2 3 , so the answer is 2 3 .
Explanation
Understanding the Problem We are given the sequence 8 , 12 , 18 , 27 , … and we need to find the common ratio in the recursive formula that represents this sequence. In a geometric sequence, the ratio between consecutive terms is constant. This constant is called the common ratio.
Calculating the Ratios To find the common ratio, we can divide any term by its preceding term. Let's calculate the ratio between the first few consecutive terms:
Ratio between the second and first terms: 8 12 Ratio between the third and second terms: 12 18 Ratio between the fourth and third terms: 18 27
Simplifying the Ratios Now, we simplify each of these fractions:
8 12 = 4 × 2 4 × 3 = 2 3 12 18 = 6 × 2 6 × 3 = 2 3 18 27 = 9 × 2 9 × 3 = 2 3
Determining the Common Ratio Since the ratio between consecutive terms is consistently 2 3 , the common ratio for the recursive formula is 2 3 .
Final Answer Therefore, the value Natalia should use as the common ratio in the formula is 2 3 .
Examples
Geometric sequences are not just abstract math; they appear in many real-world scenarios. For example, consider the growth of bacteria in a culture. If the bacteria double every hour, the number of bacteria at each hour forms a geometric sequence with a common ratio of 2. Similarly, the depreciation of a car's value each year can be modeled using a geometric sequence, where the common ratio is the percentage of the car's value retained each year. Understanding geometric sequences helps us predict and analyze these types of phenomena.
