Consider the geometric sequence:
$8,4,2,1, \ldots$
If $n$ is an integer, which of these functions generate the sequence?
Choose all answers that apply:
A $a(n)=8\left(\frac{1}{2}\right)^{n}$ for $n \geq 1$
B $b(n)=32\left(\frac{1}{2}\right)^{n}$ for $n \geq 2$
c. $c(n)=64\left(\frac{1}{2}\right)^{n}$ for $n \geq 3$

D $d(n)=128\left(\frac{1}{2}\right)^{n}$ for $n \geq 4$

Question

Consider the geometric sequence:
$8,4,2,1, \ldots$
If $n$ is an integer, which of these functions generate the sequence?
Choose all answers that apply:
A $a(n)=8\left(\frac{1}{2}\right)^{n}$ for $n \geq 1$
B $b(n)=32\left(\frac{1}{2}\right)^{n}$ for $n \geq 2$
c. $c(n)=64\left(\frac{1}{2}\right)^{n}$ for $n \geq 3$

D $d(n)=128\left(\frac{1}{2}\right)^{n}$ for $n \geq 4$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To determine which functions generate the sequence $8, 4, 2, 1, \ldots$, we need to evaluate each function at the specified values of $n$ and compare the results to the sequence. ‖ ‖ step 2 ⋮ For option A, $a(n)=8\left(\frac{1}{2}\right)^{n}$ for $n \geq 1$, we calculate the first four terms: $a(1)=8\left(\frac{1}{2}\right)^{1}=4$, $a(2)=8\left(\frac{1}{2}\right)^{2}=2$, $a(3)=8\left(\frac{1}{2}\right)^{3}=1$, and $a(4)=8\left(\frac{1}{2}\right)^{4}=\frac{1}{2}$. This does not match the sequence starting from $n=1$. ‖ ‖ step 3 ⋮ For option B, $b(n)=32\left(\frac{1}{2}\right)^{n}$ for $n \geq 2$, we calculate the first four terms starting from $n=2$: $b(2)=32\left(\frac{1}{2}\right)^{2}=8$, $b(3)=32\left(\frac{1}{2}\right)^{3}=4$, $b(4)=32\left(\frac{1}{2}\right)^{4}=2$, and $b(5)=32\left(\frac{1}{2}\right)^{5}=1$. This matches the sequence starting from $n=2$. ‖ ‖ step 4 ⋮ For option C, $c(n)=64\left(\frac{1}{2}\right)^{n}$ for $n \geq 3$, we calculate the first four terms starting from $n=3$: $c(3)=64\left(\frac{1}{2}\right)^{3}=8$, $c(4)=64\left(\frac{1}{2}\right)^{4}=4$, $c(5)=64\left(\frac{1}{2}\right)^{5}=2$, and $c(6)=64\left(\frac{1}{2}\right)^{6}=1$. This matches the sequence starting from $n=3$. ‖ ‖ step 5 ⋮ For option D, $d(n)=128\left(\frac{1}{2}\right)^{n}$ for $n \geq 4$, we calculate the first four terms starting from $n=4$: $d(4)=128\left(\frac{1}{2}\right)^{4}=8$, $d(5)=128\left(\frac{1}{2}\right)^{5}=4$, $d(6)=128\left(\frac{1}{2}\right)^{6}=2$, and $d(7)=128\left(\frac{1}{2}\right)^{7}=1$. This matches the sequence starting from $n=4$. ‖ ***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 $n$. Options B, C, and D generate the sequence $8, 4, 2, 1, \ldots$ when starting at their respective values of $n$.⚹◊Which function accurately generates the geometric sequence $8, 4, 2, 1, ...$ for the given values of $n$?⍭ Generate me a similar question◊

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