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A sequence is defined recursively using the formula [tex]f(n+1) = -0.5 f(n)[/tex]. If the first term of the sequence is 120, what is [tex]f(5)[/tex]?

A. [tex]\(-15\)[/tex]
B. [tex]\(-7.5\)[/tex]
C. [tex]\(7.5\)[/tex]
D. [tex]\(15\)[/tex]

Sagot :

GinnyAnsweridn
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]
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