Which recursive formula can be used to generate the sequence shown, where f(1) = 9.6 and n > 1?

Question

9.6, –4.8, 2.4, –1.2, 0.6, ...

f(n + 1) = (–0.5)f(n)
f(n + 1) = (0.5)f(n)
f(n + 1) = f(0.5n)
f(n + 1) = f(–0.5n)

Answer 1

The given sequence is: \(9.6, -4.8, 2.4, -1.2, 0.6, \ldots\).

Let's examine how the terms change from one to the next:
- Second term \(-4.8\) is obtained by multiplying the first term \(9.6\) by \(-0.5\): \(9.6 \times -0.5 = -4.8\).
- Third term \(2.4\) is obtained by multiplying the second term \(-4.8\) by \(-0.5\): \(-4.8 \times -0.5 = 2.4\).
- Fourth term \(-1.2\) is obtained by multiplying the third term \(2.4\) by \(-0.5\): \(2.4 \times -0.5 = -1.2\).
- Fifth term \(0.6\) is obtained by multiplying the fourth term \(-1.2\) by \(-0.5\): \(-1.2 \times -0.5 = 0.6\).

From this pattern, we observe that each term is derived by multiplying the previous term by \(-0.5\).

Therefore, the recursive formula that can be used to generate this sequence is:
\[ f(n + 1) = (-0.5)f(n) \]