Question

1. U.S. Population. An arithmetic sequence is the discrete analog to a linear function. Both an arithmetic sequence and a linear function assumption assume a constant additive rate of change.
a) According to the U.S. Census Bureau, the official U.S. population was 248.7 million in 1990 and 308.7 million in 2010. What was the average change in population per year from 1990 through 2010?

(308.7 mil-248.7 mil)/(2010-1990)
= 60mil/20 years
= 3mil/per year

The official U.S. population was 281.4 million in 2000. Let pn be an arithmetic sequence that represents the U.S. population (in millions of persons) n years after 2000. p0 = 281.4. This is the intial condition for the sequence.


b) Use your answer from part a to write a recursive formula for the sequence pn.

c) Write an explicit formula for the arithmetic sequence pn, assuming that
p0 = 281.4.
d) Use the information from parts b, c, and d to estimate the U.S. population in 2010, 2015, and 2020.

Answer 1

A.) Given that the official U.S. population was 248.7 million in 1990 and 308.7 million in 2010.

To find the average change in population per year from 1990 through 2010, let the year be represented by x, and the population in that year be represented by f(x).

Recall that the average rate of change of a function over a given interval, (a, b) is given by
$\frac{f(b)-f(a)}{b-a}$

Thus, the average change in population per year from 1990 through 2010 is given by
$\frac{308.7-248.7}{2010-1990} = \frac{60}{20} =3$

Therefore, the average change in population per year from 1990 through 2010 is 3 million.


B.) From part a, we obtained that the population grows at a rate of 3 million per year.
Given that $p_0=281.4$ is the initial value of the population in 2000 and $p_n$ is the population, n, years after 2000.

Thus, a recursive formula for the sequence $p_n$ is given by
$p_n=p_{n-1}+3, \ p_0=281.4$


C.) Recall that the explicit formula for an arithmetic sequence with firstterm, a, and common difference, d is given by
$p_n=a+(n-1)d$

Given that a = $p_0$ = 281.4 and, d = 3, thus an explicit formula for the arithmetic sequence $p_n$ is given by
$p_n=281.4+(n-1)3=281.4+3n-3=278.4+3n \\ \\ \bold{p_n=278.4+3n}$


D.) Using the formula obtained in part C, notice that 2010 is 10 years after 2000, 2015 is 15 years and 2020 is 20 years after 2000.
For, 2010:
$p_n=278.4+3(10)=278.4+30=308.4$
For 2015:
$p_n=278.4+3(15)=278.4+45=323.4$
For 2020:
$p_n=278.4+3(20)=278.4+60=338.4$