Question

8.

n	f(n)
2	24
3
4
5	-648

explicit equation: f(n) = f(1)·r^(n-1) → f(n) = f(1)·r^(n-1)
recursive equation: f(n) = f(n - 1)·r → f(n) = f(n - 1)·r

Explanation:

Step1: Identify the sequence type

The explicit formula \( f(n) = f(1) \cdot r^{n - 1} \) and recursive formula \( f(n)=f(n - 1)\cdot r \) indicate a geometric sequence, where \( f(1) \) is the first term and \( r \) is the common ratio.

Step2: Use the given terms to find \( r \) and \( f(1) \)

We know \( f(2) = 24 \) and \( f(5)=- 648 \). From the explicit formula:

• For \( n = 2 \): \( f(2)=f(1)\cdot r^{2 - 1}=f(1)\cdot r=24\) --- (1)
• For \( n = 5 \): \( f(5)=f(1)\cdot r^{5 - 1}=f(1)\cdot r^{4}=-648\) --- (2)

Divide equation (2) by equation (1):

$$ \frac{f(1)\cdot r^{4}}{f(1)\cdot r}=\frac{-648}{24} $$

Simplify the left - hand side: \( r^{3}\), and the right - hand side: \(-27\)
So \( r^{3}=-27\), then \( r = \sqrt[3]{-27}=-3\)

Step3: Find \( f(1) \)

Substitute \( r=-3 \) into equation (1): \( f(1)\cdot(-3)=24\), so \( f(1)=\frac{24}{-3}=- 8\)

Step4: Find the missing terms

• For \( n = 1 \): \( f(1)=-8\)
• For \( n = 3 \): Using the recursive formula \( f(3)=f(2)\cdot r\). Since \( f(2) = 24 \) and \( r=-3 \), then \( f(3)=24\times(-3)=-72\)
• For \( n = 4 \): Using the recursive formula \( f(4)=f(3)\cdot r\). Since \( f(3)=-72 \) and \( r = - 3\), then \( f(4)=-72\times(-3)=216\)

Answer:

• \( f(1)=-8 \)
• \( f(3)=-72 \)
• \( f(4)=216 \)

The completed table (filling in the missing \( f(n) \) values) is:

\( n \)	\( 1 \)	\( 2 \)	\( 3 \)	\( 4 \)	\( 5 \)