c Use a spreadsheet and the recursive rule from Part A to calculate more terms of the sequence.
A
aRecursive Rule: a_1=50 000, a_n=0.8a_(n-1)+5000 for n>1
B
b 35 240
C
c The number of members stabilizes at about 25 000 people.
Practice makes perfect
a Let a_n be the number of members at the beginning of the n^(th) year. The music service initially has 50 000 members. This tells us that a_1=50 000. Now, we will find a recursive equation for the sequence.
Recursive Equation
At the beginning of the (n-1)^(th) year the number of members is a_(n-1). Each year, the company loses 20 %= 0.2 of members. The number of old members at the end of (n-1)^(th) year is (1- 0.2)a_(n-1)= 0.8a_(n-1). The service gains 5000 new members every year. This tells us that a_n= 0.8a_(n-1)+ 5000.
Recursive Rule:
a_1=50 000, a_n=0.8a_(n-1)+5000forn>1
b We are asked to find a_5. To do this we will use a spreadsheet. Let's enter a_1=50 000 in cell A1 and write =Round((0.8)*(A1)+5000,2) in cell A2. When we hit enter, we will get the following.
Next, copy cell A2, highlight cells A3 through A5, and paste.
We get 35 240 in the fifth row. This tells us that a_5=35 240.
c We are asked to find what happens to the number of members over time. To do this, we will use a spreadsheet. Let's enter a_1=50 000 in cell A1 and write =Round(0.8)*(A1)+5000 in cell A2. When we hit enter, we will get the following.
Next, copy cell A2, highlight cells A3 through A100, and paste.
Notice that the sequence stabilizes around cell A63.
After cell A62 we will keep getting the same value, 25 000.02. This tells us that the number of members stabilizes around 25 000 people.
