Question

which formula can be used to describe the sequence?
64, -48, 36, -27, …
$f(x + 1) = -\frac{3}{4}f(x)$
$f(x) = \frac{3}{4}f(x + 1)$
$f(x) = -\frac{3}{4}f(x + 1)$

Explanation:

Step1: Find the common ratio

To find the common ratio \( r \) of the geometric sequence, divide a term by its previous term. Let's take the second term \(-48\) and divide by the first term \(64\):
\( r = \frac{-48}{64} = -\frac{3}{4} \).

This means each term \( f(x + 1) \) is \( -\frac{3}{4} \) times the previous term \( f(x) \), so \( f(x + 1) = -\frac{3}{4}f(x) \).

We can also verify with other terms: \(\frac{36}{-48} = -\frac{3}{4}\) and \(\frac{-27}{36} = -\frac{3}{4}\), confirming the common ratio is \( -\frac{3}{4} \).

Now, let's check the other options:

• For \( f(x) = \frac{3}{4}f(x + 1) \), rearranging gives \( f(x + 1) = \frac{4}{3}f(x) \), which does not match the common ratio.
• For \( f(x) = -\frac{3}{4}f(x + 1) \), rearranging gives \( f(x + 1) = -\frac{4}{3}f(x) \), which also does not match the common ratio.

Answer:

\( f(x + 1) = -\frac{3}{4}f(x) \) (the first option)