c Use a spreadsheet and the recursive rule from Part A to calculate more terms of the sequence.
A
aRecursive Rule: a_1=34, a_n=0.6a_(n-1)+34 for n>1
B
b 37.84
C
c The amount of chlorine stabilizes at about 40 ounces.
Practice makes perfect
a Let a_n be the amount of ounces of chlorine in a swimming pool at the beginning of the n^(th) week. In the first week we add 34 ounces of chlorine. This tells us that a_1=34. Now, we will find a recursive equation for the sequence.
Recursive Equation
At the beginning of the (n-1)^(th) week the amount of chlorine is a_(n-1). Each week, the pool loses 40 %= 0.4 of chlorine. The amount of old chlorine at the end of the (n-1)^(th) week is (1- 0.4)a_(n-1)= 0.6a_(n-1). We add 16 ounces of chlorine every week. This tells us that a_n= 0.6a_(n-1)+ 16.
Recursive Rule:
a_1=34, a_n=0.6a_(n-1)+16forn>1
b We are asked to find a_3. To do this, we will use a spreadsheet. Let's enter a_1=34 in cell A1 and write =Round((0.6)*(A1)+16,2) in cell A2. When we hit enter, we will get the following.
Next, copy cell A2, highlight cell A3, and paste.
We get a number 37.84 in the third row. This tells us that a_3=37.84.
c We are asked to find what happens to the amount of chlorine over time. To do this, we will use a spreadsheet. Let's enter a1=34 in cell A1 and write =Round(0.6)*(A1)+16 in cell A2. When we hit enter, we will get the following.
Next, copy cell A2, highlight cells A3 through A14, and paste.
After cell A13 we will keep getting the same value, 39.99. This tells us that the amount of chlorine stabilizes around 40 ounces.
