Question

1. write an explicit and recursive rule for the geometric sequence 2, -6, 18, -54, ...

$f(1) =$
$r =$
explicit: $f(n) =$
recursive: $f(n) =$
$f(1) =$

Explanation:

Step1: Identify first term

$f(1) = 2$

Step2: Calculate common ratio

$r = \frac{-6}{2} = -3$

Step3: Write explicit formula

Use geometric sequence form $f(n)=f(1)\cdot r^{n-1}$
$f(n) = 2\cdot (-3)^{n-1}$

Step4: Write recursive formula

Recursive rule uses prior term: $f(n)=r\cdot f(n-1)$, with initial term
$f(n) = -3\cdot f(n-1)$
$f(1) = 2$

Answer:

$f(1) = 2$
$r = -3$
Explicit: $f(n) = 2\cdot (-3)^{n-1}$
Recursive: $f(n) = -3\cdot f(n-1)$
$f(1) = 2$