In 1985, 10% of Americans had no close friends. On average, this has increased by approximately 0.77% per year
a. Write a formula for the nth term of the arithmetic sequence that models the percentage of Americans with no close friends n years after 1984.
b. If trends shown by the model in part (a) continue, what percentage of Americans will have no close friends in 2011?
you have a sequence where
a = 0.1
r = 1.0077
Now you can write the formulas needed.
a. To find the formula for the nth term of an arithmetic sequence, we need to first identify the first term (a) and the common difference (d).
In this case, the first term (a) is 10% since 10% of Americans had no close friends in 1985.
The common difference (d) is 0.77% since the percentage increased by approximately 0.77% per year.
The formula for the nth term of an arithmetic sequence is:
an = a + (n - 1)d
Substituting the values we have:
an = 10% + (n - 1) * 0.77%
Therefore, the formula for the nth term of this arithmetic sequence is an = 10% + 0.77%(n - 1).
b. To determine the percentage of Americans with no close friends in 2011, we need to calculate the value of an when n = 2011 - 1984.
First, calculate n - 1:
n - 1 = 2011 - 1984 - 1 = 26
Now, substitute this value into the formula:
an = 10% + 0.77%(n - 1)
an = 10% + 0.77%(26)
an = 10% + 0.77% * 26
Calculating 0.77% * 26:
0.77% * 26 = 0.2
Adding to 10%:
an = 10% + 0.2
an = 10.2%
Therefore, if the trends shown by the model continue, approximately 10.2% of Americans will have no close friends in 2011.