Question

write a function to represent the geometric sequence 48, 36, 27, ... (1 point) \\(\bigcirc\\) \\(f(x) = 36(0.75)^{x - 1}\\) \\(\bigcirc\\) \\(f(x) = 48(0.75)^{x - 1}\\) \\(\bigcirc\\) \\(f(x) = 36 - (0.75)(x - 1)\\) \\(\bigcirc\\) \\(f(x) = 48(1.33)^{x - 1}\\)

Explanation:

Step1: Recall geometric sequence formula

The general formula for a geometric sequence is \( f(x)=a(r)^{x - 1} \), where \( a \) is the first term and \( r \) is the common ratio.

Step2: Identify the first term (\(a\))

In the sequence \( 48, 36, 27, \dots \), the first term \( a = 48 \).

Step3: Calculate the common ratio (\(r\))

The common ratio \( r=\frac{\text{second term}}{\text{first term}}=\frac{36}{48} = 0.75 \).

Step4: Substitute \(a\) and \(r\) into the formula

Substituting \( a = 48 \) and \( r=0.75 \) into \( f(x)=a(r)^{x - 1} \), we get \( f(x)=48(0.75)^{x - 1} \). Also, we can eliminate the third option as it is an arithmetic sequence - like formula (subtraction) and the fourth option has a ratio greater than 1 (but our ratio is 0.75) and the first option has the wrong first term (36 instead of 48).

Answer:

\( f(x) = 48(0.75)^{x - 1} \) (the second option)