To find the fifth term [tex]\( f(5) \)[/tex] in the sequence defined by the recursive formula [tex]\( f(n+1) = -0.5 f(n) \)[/tex], with the initial term [tex]\( f(1) = 120 \)[/tex], we need to calculate each term step-by-step.
Given:
[tex]\[ f(1) = 120 \][/tex]
We will apply the recursive formula to determine the subsequent terms.
1. Calculate [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = -0.5 \times f(1) = -0.5 \times 120 = -60 \][/tex]
2. Calculate [tex]\( f(3) \)[/tex]:
[tex]\[ f(3) = -0.5 \times f(2) = -0.5 \times -60 = 30 \][/tex]
3. Calculate [tex]\( f(4) \)[/tex]:
[tex]\[ f(4) = -0.5 \times f(3) = -0.5 \times 30 = -15 \][/tex]
4. Calculate [tex]\( f(5) \)[/tex]:
[tex]\[ f(5) = -0.5 \times f(4) = -0.5 \times -15 = 7.5 \][/tex]
Thus, the fifth term [tex]\( f(5) \)[/tex] in the sequence is [tex]\( 7.5 \)[/tex]. Therefore, the correct answer is:
[tex]\[ 7.5 \][/tex]
