Question

geometric sequences
determining a term in a geometric sequence given the initial value and recursive formula
what is $f(5)$ if $f(1) = 3.2$ and $f(x + 1) = \frac{5}{2}(f(x))$?

Explanation:

Step1: Find \( f(2) \)

Given \( f(1) = 3.2 \) and \( f(x + 1)=\frac{5}{2}(f(x)) \), substitute \( x = 1 \):
\( f(2)=\frac{5}{2}(f(1))=\frac{5}{2}(3.2)=8 \)

Step2: Find \( f(3) \)

Substitute \( x = 2 \) into the recursive formula:
\( f(3)=\frac{5}{2}(f(2))=\frac{5}{2}(8)=20 \)

Step3: Find \( f(4) \)

Substitute \( x = 3 \) into the recursive formula:
\( f(4)=\frac{5}{2}(f(3))=\frac{5}{2}(20)=50 \)

Step4: Find \( f(5) \)

Substitute \( x = 4 \) into the recursive formula:
\( f(5)=\frac{5}{2}(f(4))=\frac{5}{2}(50)=125 \)

Answer:

\( 125 \)