Answer :
To find the fifth term [tex]\( f(5) \)[/tex] in the given sequence where [tex]\( f(n+1) = -0.5 f(n) \)[/tex] and [tex]\( f(0) = 120 \)[/tex], we can proceed step-by-step starting from [tex]\( f(0) \)[/tex].
1. First Term [tex]\((f(0))\)[/tex]:
[tex]\[ f(0) = 120 \][/tex]
2. Second Term [tex]\((f(1))\)[/tex]:
[tex]\[ f(1) = -0.5 \cdot f(0) = -0.5 \cdot 120 = -60 \][/tex]
3. Third Term [tex]\((f(2))\)[/tex]:
[tex]\[ f(2) = -0.5 \cdot f(1) = -0.5 \cdot -60 = 30 \][/tex]
4. Fourth Term [tex]\((f(3))\)[/tex]:
[tex]\[ f(3) = -0.5 \cdot f(2) = -0.5 \cdot 30 = -15 \][/tex]
5. Fifth Term [tex]\((f(4))\)[/tex]:
[tex]\[ f(4) = -0.5 \cdot f(3) = -0.5 \cdot -15 = 7.5 \][/tex]
6. Sixth Term [tex]\((f(5))\)[/tex]:
[tex]\[ f(5) = -0.5 \cdot f(4) = -0.5 \cdot 7.5 = -3.75 \][/tex]
Thus, the fifth term [tex]\( f(5) \)[/tex] in the sequence is:
[tex]\[ f(5) = -3.75 \][/tex]
This does not match any of the provided options exactly, which means there might have been an error in the provided answer choices. However, based on our step-by-step calculation, the correct answer should be:
[tex]\[ -3.75 \][/tex]
1. First Term [tex]\((f(0))\)[/tex]:
[tex]\[ f(0) = 120 \][/tex]
2. Second Term [tex]\((f(1))\)[/tex]:
[tex]\[ f(1) = -0.5 \cdot f(0) = -0.5 \cdot 120 = -60 \][/tex]
3. Third Term [tex]\((f(2))\)[/tex]:
[tex]\[ f(2) = -0.5 \cdot f(1) = -0.5 \cdot -60 = 30 \][/tex]
4. Fourth Term [tex]\((f(3))\)[/tex]:
[tex]\[ f(3) = -0.5 \cdot f(2) = -0.5 \cdot 30 = -15 \][/tex]
5. Fifth Term [tex]\((f(4))\)[/tex]:
[tex]\[ f(4) = -0.5 \cdot f(3) = -0.5 \cdot -15 = 7.5 \][/tex]
6. Sixth Term [tex]\((f(5))\)[/tex]:
[tex]\[ f(5) = -0.5 \cdot f(4) = -0.5 \cdot 7.5 = -3.75 \][/tex]
Thus, the fifth term [tex]\( f(5) \)[/tex] in the sequence is:
[tex]\[ f(5) = -3.75 \][/tex]
This does not match any of the provided options exactly, which means there might have been an error in the provided answer choices. However, based on our step-by-step calculation, the correct answer should be:
[tex]\[ -3.75 \][/tex]
