Question

which function can be used to model the graphed geometric sequence? the graph has points (1, 1296), (2, 1080), (3, 900), (4, 750), (5, 625). the options are: $f(x + 1) = \frac{6}{5}f(x)$; $f(x + 1) = \frac{5}{6}f(x)$; $f(x + 1) = \frac{5}{6}f(x)$ (and another option partially visible)

Explanation:

Step1: Find the common ratio

To find the common ratio \( r \) of a geometric sequence, we divide a term by its previous term. Let's take the second term \( f(2) = 1080 \) and the first term \( f(1) = 1296 \). So \( r=\frac{f(2)}{f(1)}=\frac{1080}{1296}\). Simplify this fraction: divide numerator and denominator by 216, we get \( \frac{1080\div216}{1296\div216}=\frac{5}{6} \). Let's check with another pair, say \( f(3) = 900 \) and \( f(2) = 1080 \). \( \frac{900}{1080}=\frac{5}{6} \) (divide numerator and denominator by 180). So the common ratio \( r = \frac{5}{6} \).

Step2: Relate to the recursive formula

In a geometric sequence, the recursive formula is \( f(x + 1)=r\times f(x) \). Since we found \( r=\frac{5}{6} \), the formula should be \( f(x + 1)=\frac{5}{6}f(x) \).

Answer:

\( f(x + 1)=\frac{5}{6}f(x) \) (the option with \( f(x + 1)=\frac{5}{6}f(x) \))