Question

consider this geometric sequence:

(4, -12, 36, -108, \ldots)

what is the explicit function that defines this sequence?

enter the correct answer in the box. replace the (a) and (r) terms with the correct values.

Explanation:

🆕 New Concept Discovered: Geometric Sequences
Finding explicit formulas using the first term and common ratio

Step 1: Identify the first term

A geometric sequence has an explicit formula of the form:

$$ f(n) = a \cdot r^{n-1} $$

where \( a \) is the first term and \( r \) is the common ratio.

Looking at the given sequence:

$$ 4, -12, 36, -108, \dots $$

The first term \( a \) is:

$$ a = 4 $$

Step 2: Find the common ratio

To find the common ratio \( r \), divide any term by the preceding term:

$$ r = \frac{-12}{4} = -3 $$

We can verify this with the next term:

$$ \frac{36}{-12} = -3 $$

So, the common ratio \( r \) is \( -3 \).

Step 3: Write the explicit function

Substitute \( a = 4 \) and \( r = -3 \) into the standard explicit formula \( f(n) = a \cdot r^{n-1} \):

$$ f(n) = 4 \cdot (-3)^{n-1} $$

Answer:

$$ f(n) = 4 \cdot (-3)^{n-1} $$