Question

1. a piece of paper has an area of 256 cm². a strip is cut off that is 1/4 the original area. from that strip, another strip is cut off that is 1/4 the last strip, and so on. below is a graph and table representing sequence p, where p(n) is the area in square - centimeters of the strip of paper after n cuts.

a) (i) is sequence p geometric or arithmetic? (ii) explain how you know.

b) write an equation to define sequence p recursively. include the n ≥ statement.

c) what is a reasonable domain for p(n)? that is, what range of values for the number of cuts actually make sense in the context? what are some values of n (the number of cuts) that dont make sense to use? explain your reasoning.

i. domain (all values that make sense for the number of cuts)

why are the number of cuts limited in the domain?

ii. values that dont make sense for the number of cuts

explain why not.

Explanation:

Step1: Determine the nature of the sequence

A geometric sequence has a common - ratio between consecutive terms. Given $p(0) = 256$, $p(1)=64$, $p(2) = 16$, $p(3)=4$, $p(4)=1$. The ratio $\frac{p(1)}{p(0)}=\frac{64}{256}=\frac{1}{4}$, $\frac{p(2)}{p(1)}=\frac{16}{64}=\frac{1}{4}$, $\frac{p(3)}{p(2)}=\frac{4}{16}=\frac{1}{4}$, $\frac{p(4)}{p(3)}=\frac{1}{4}$. Since the ratio between consecutive terms is constant ($\frac{1}{4}$), the sequence is geometric.

Step2: Write the recursive formula

For a geometric sequence, the recursive formula is $p(n)=r\times p(n - 1)$, where $r$ is the common - ratio. Here, $r=\frac{1}{4}$, so $p(n)=\frac{1}{4}p(n - 1)$ for $n\geq1$ and $p(0)=256$.

Step3: Find the domain

The number of cuts $n$ must be a non - negative integer. You can't have a negative number of cuts or a non - integer number of cuts (like half a cut) in the context of physically cutting a piece of paper. So the domain of $p(n)$ is the set of non - negative integers $\{n\in\mathbb{Z}:n\geq0\}$. Values that don't make sense are negative integers (e.g., $n=-1$) because you can't have a negative number of cuts, and non - integer values (e.g., $n = 0.5$) because you can't make a fraction of a cut.

Answer:

a) The sequence $p$ is geometric because the ratio between consecutive terms is constant ($\frac{1}{4}$).
b) $p(n)=\frac{1}{4}p(n - 1)$ for $n\geq1$, $p(0)=256$.
c)
I. Domain: The set of non - negative integers $\{n\in\mathbb{Z}:n\geq0\}$. Reason: The number of cuts must be a whole number and cannot be negative.
II. Values that don't make sense: Negative integers and non - integer values. Reason: You can't have a negative number of cuts or a fraction of a cut when physically cutting a piece of paper.