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Question
Math
Posted 3 months ago
Consider the geometric sequence:
8,4,2,1,8,4,2,1, \ldots
If nn is an integer, which of these functions generate the sequence?
Choose all answers that apply:
A a(n)=8(12)na(n)=8\left(\frac{1}{2}\right)^{n} for n1n \geq 1
B b(n)=32(12)nb(n)=32\left(\frac{1}{2}\right)^{n} for n2n \geq 2
c. c(n)=64(12)nc(n)=64\left(\frac{1}{2}\right)^{n} for n3n \geq 3

D d(n)=128(12)nd(n)=128\left(\frac{1}{2}\right)^{n} for n4n \geq 4
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Answer from Sia
Posted 3 months ago
Solution by Steps
step 2
For option A, a(n)=8(12)na(n)=8\left(\frac{1}{2}\right)^{n} for n1n \geq 1, we calculate the first four terms: a(1)=8(12)1=4a(1)=8\left(\frac{1}{2}\right)^{1}=4, a(2)=8(12)2=2a(2)=8\left(\frac{1}{2}\right)^{2}=2, a(3)=8(12)3=1a(3)=8\left(\frac{1}{2}\right)^{3}=1, and a(4)=8(12)4=12a(4)=8\left(\frac{1}{2}\right)^{4}=\frac{1}{2}. This does not match the sequence starting from n=1n=1
step 3
For option B, b(n)=32(12)nb(n)=32\left(\frac{1}{2}\right)^{n} for n2n \geq 2, we calculate the first four terms starting from n=2n=2: b(2)=32(12)2=8b(2)=32\left(\frac{1}{2}\right)^{2}=8, b(3)=32(12)3=4b(3)=32\left(\frac{1}{2}\right)^{3}=4, b(4)=32(12)4=2b(4)=32\left(\frac{1}{2}\right)^{4}=2, and b(5)=32(12)5=1b(5)=32\left(\frac{1}{2}\right)^{5}=1. This matches the sequence starting from n=2n=2
step 4
For option C, c(n)=64(12)nc(n)=64\left(\frac{1}{2}\right)^{n} for n3n \geq 3, we calculate the first four terms starting from n=3n=3: c(3)=64(12)3=8c(3)=64\left(\frac{1}{2}\right)^{3}=8, c(4)=64(12)4=4c(4)=64\left(\frac{1}{2}\right)^{4}=4, c(5)=64(12)5=2c(5)=64\left(\frac{1}{2}\right)^{5}=2, and c(6)=64(12)6=1c(6)=64\left(\frac{1}{2}\right)^{6}=1. This matches the sequence starting from n=3n=3
step 5
For option D, d(n)=128(12)nd(n)=128\left(\frac{1}{2}\right)^{n} for n4n \geq 4, we calculate the first four terms starting from n=4n=4: d(4)=128(12)4=8d(4)=128\left(\frac{1}{2}\right)^{4}=8, d(5)=128(12)5=4d(5)=128\left(\frac{1}{2}\right)^{5}=4, d(6)=128(12)6=2d(6)=128\left(\frac{1}{2}\right)^{6}=2, and d(7)=128(12)7=1d(7)=128\left(\frac{1}{2}\right)^{7}=1. This matches the sequence starting from n=4n=4
B, C, D
Key Concept
Geometric Sequence and Exponential Functions
Explanation
The correct functions must generate the terms of the geometric sequence when evaluated at the specified values of nn. Options B, C, and D generate the sequence 8,4,2,1,8, 4, 2, 1, \ldots when starting at their respective values of nn.

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