Which recursive formula can be used to generate the sequence shown, where f(1)=9.6 and n≥1 ? 9.6,-4.8,2.4,-1.2,0.6,dots f(n+1)=(-0.5)f(n) f(n+1)=(0.5)(n) f(n+1)=f(0.5n) f(n+1)=
The Correct Answer and Explanation is:
To find the recursive formula for the sequence given, where ( f(1) = 9.6 ) and the sequence continues as ( 9.6, -4.8, 2.4, -1.2, 0.6, \dots ), we need to identify how each term relates to the previous term.
Analysis of the Sequence
Let’s denote the sequence by ( f(n) ):
- ( f(1) = 9.6 )
- ( f(2) = -4.8 )
- ( f(3) = 2.4 )
- ( f(4) = -1.2 )
- ( f(5) = 0.6 )
Now, we can observe the relationship between consecutive terms:
- From ( f(1) ) to ( f(2) ):
[
f(2) = -0.5 \cdot f(1) = -0.5 \cdot 9.6 = -4.8
] - From ( f(2) ) to ( f(3) ):
[
f(3) = -0.5 \cdot f(2) = -0.5 \cdot (-4.8) = 2.4
] - From ( f(3) ) to ( f(4) ):
[
f(4) = -0.5 \cdot f(3) = -0.5 \cdot 2.4 = -1.2
] - From ( f(4) ) to ( f(5) ):
[
f(5) = -0.5 \cdot f(4) = -0.5 \cdot (-1.2) = 0.6
]
This consistent pattern shows that each term ( f(n+1) ) is generated by multiplying the previous term ( f(n) ) by (-0.5).
Recursive Formula
Thus, the correct recursive formula for this sequence is:
[
f(n+1) = -0.5 \cdot f(n)
]
for ( n \geq 1 ).
Explanation
This recursive relationship is characteristic of geometric sequences, where each term is derived by multiplying the previous term by a constant ratio. In this case, the ratio is (-0.5), which explains the alternating signs and the gradual reduction in magnitude of the sequence.
The sequence starts with a positive value, and as it progresses, the terms decrease in absolute value and switch between positive and negative due to the negative multiplier. The sequence is bounded, and we can see that it converges toward zero as ( n ) increases.
The formula ( f(n+1) = -0.5 \cdot f(n) ) captures the essence of the sequence’s behavior and allows us to compute any term in the sequence recursively from the previous term, starting from the initial value of ( 9.6 ).