Questions: Consider the geometric sequence below. -21, 63, -189, 567, ... Complete the recursively defined function to describe this sequence. f(1)= f(n)= f(n-1) for n=2,3,4...

Consider the geometric sequence below.
-21, 63, -189, 567, ...

Complete the recursively defined function to describe this sequence.
f(1)=
f(n)= f(n-1) for n=2,3,4...

Transcript text: Consider the geometric sequence below. \[ -21,63,-189,567, \ldots \] Complete the recursively defined function to describe this sequence. \[ f(1)= \] $f(n)=$ $\square$ $f(n-1)$ for $n=2,3,4 \ldots$

Solution

Solution Steps

Step 1: Identify the first term of the sequence

The first term of the geometric sequence is given as $ f(1) = -21 $.

Step 2: Determine the common ratio

To find the common ratio $ r $, divide the second term by the first term: \[ r = \frac{63}{-21} = -3 \] The common ratio is $ r = -3 $.

Step 3: Write the recursive formula

The recursive formula for a geometric sequence is: \[ f(n) = r \cdot f(n-1) \] Substituting the common ratio $ r = -3 $, the recursive formula becomes: \[ f(n) = -3 \cdot f(n-1) \]

Final Answer

\[ f(1) = -21, \quad f(n) = -3 \cdot f(n-1) \text{ for } n=2,3,4 \ldots \]

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