Question

find an explicit formula, f(n), for the following geometric sequence: 2, -6, 18, -54, ...

Explanation:

Step1: Identify first term $a_1$

The first term of the sequence is $a_1 = 2$.

Step2: Calculate common ratio $r$

Divide any term by the prior term: $r = \frac{-6}{2} = -3$.

Step3: Apply geometric sequence formula

The explicit formula for a geometric sequence is $f(n) = a_1 \cdot r^{n-1}$. Substitute $a_1=2$ and $r=-3$:
$f(n) = 2 \cdot (-3)^{n-1}$

Answer:

$f(n) = 2 \cdot (-3)^{n-1}$