Question

A piece of paper has an area of 81 cm². A strip is cut off that is 1/3 the original area. From that strip, another strip is cut off that is 1/3 the area of the first, and so on. Here is a graph and table representing sequence k, where k(n) is the area in square centimeters of the strip of paper after n cuts.

0 area in square centimeters 80 75 70 65 60 55 50 45 40 . 25 20 15 10 5 2 3 number of cuts number area in square of cuts centimeters 0 81 1 27 2 9 3 3 4 1
a) Is sequence k geometric or arithmetic? Explain how you know.
b) Write an equation to define sequence k recursively.
c) For term k(n), what are some values of n that make sense to use? What are some values of n that don't make sense to use? Explain your reasoning. (Hint: Think about positive numbers, negative numbers, decimals, and 0)

Answer 1

The sequence in the scenario is geometric because each term is 1/3 of the previous one. The recursive equation is k(n) = k(n-1) × (1/3), where k(0) = 81. Values for n are non-negative integers, as they represent the number of cuts made.

Step-by-step explanation:

The sequence described in the scenario is a geometric sequence because each term is obtained by multiplying the previous term by a constant ratio. In this case, the ratio is 1/3 as each subsequent strip cut off is one-third of the area of the strip before it. We know this since the area of the strip after the first cut is one-third of the original area (81 cm²), and the pattern continues with each cut.

The recursive equation for sequence k can be defined as follows: k(n) = k(n-1) × (1/3), where k(0) = 81, representing the area before any cuts are made.

For term k(n), integer values of n ≥ 0 make sense because you can't have a negative number of cuts, nor can you cut a fractional part of a strip if following the original cutting pattern. Thus, only non-negative integers are meaningful in this context, as they represent the number of complete cuts made.