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.

1. the first two numbers in a sequence k are k(1)=3 and k(2)=15.

a) if k is an arithmetic sequence, write a definition for the nth term of k. include the n≥ statement.

explain or show your reasoning for the expression of the nth - term definition. for example, why did you use the numbers and variable expression in the way that you did?

b) if k is a geometric sequence, write a definition for the nth term of k. include the n≥ statement.

explain or show your reasoning for the expression of the nth - term definition. for example, why did you use the numbers and variable expression in the way that you did?

Explanation:

Step1: Determine sequence type

For sequence \(p\), check the ratios of consecutive terms. \(\frac{64}{256}=\frac{1}{4}\), \(\frac{16}{64}=\frac{1}{4}\), \(\frac{4}{16}=\frac{1}{4}\), \(\frac{1}{4}=\frac{1}{4}\). Since the ratio between consecutive terms is constant (\(\frac{1}{4}\)), it is a geometric sequence.

Step2: Write recursive formula

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

Step3: Find domain

The number of cuts \(n\) must be a non - negative integer. So the domain of \(p(n)\) is \(n\in\{0,1,2,\cdots\}\) (the set of all non - negative integers). Negative values of \(n\) don't make sense because you can't have a negative number of cuts. Non - integer values of \(n\) don't make sense in the context of making cuts as you can only make a whole number of cuts.

Answer:

a) (i) Geometric
(ii) The ratio between consecutive terms is constant (\(\frac{1}{4}\)).
b) \(p(n)=\frac{1}{4}p(n - 1)\), for \(n\geq1\) and \(p(0)=256\)
c) Domain: \(n\in\{0,1,2,\cdots\}\)
Values that don't make sense: Negative values and non - integer values.
Reason: You can't have a negative number of cuts and you can only make a whole number of cuts.