Section: 11.2 / Arithmetic Sequences / Semester 2 - Day 37
Arithmetic Sequences
The bar graph in the figure to the right shows the annual salaries, rounded to the nearest thousand dollars, of U.S. senators from 2000 to 2005. The graph illustrates that each year, the salaries increased by $4 thousand.
The sequence 142, 146, 150, 154, 158, 162, … shows that each terms after the first differs from thepreceding terms by a constant, namely 4. This sequence is called an arithmetic sequence.
Activity: Find the common difference for each arithmetic sequence.
a. -5, -2, 1, 4, 7, …b. 8, 3, -2, -7, -12, …
c. If you were to sketch a graph of either series in part (a) or (b), what would you notice?
Writing the Terms of an Arithmetic Sequence
Write the first six terms of the arithmetic sequence in which
The General Term of an Arithmetic Series
If we were to build the first six terms of an arithmetic sequence whose first term is and whose common difference is d, we would have the following:
Using the Formula for the General Term of an Arithmetic Sequence
Find the ninth term of the arithmetic sequence whose first term is 6 and whose common
difference is -5.
Using an Arithmetic Sequence to Model “Meals Behind the Wheel”
According the Newsweek Magazine, thanks to drive-thrus and curbside delivery, Americans are eating more meals behind the wheel. In 2004, we averaged 32 á la car meals annually, increasing by approximately 0.7 meal per year.
a. Write a formula for the nth term of the arithmetic sequence that models the average number of car meals n
years after 2003.
b. How many car meals will the average American eat in the year 2014?
The Sum of the First n Terms if an Arithemtic Sequence
The sum of the first n terms of an arithmetic sequence, denoted by , and called the nth partial sum, can be found without having to add up all the terms. This is especially nice when we have many terms in our sequence.
PROOF:
The above proof was replicated by the German mathematician Carl Friedrich Gauss when he was 5 years old.
Using Snto Evaluate a Summation
Find the following sum: .
Modeling Total Residential Community Costs over a Six-Year Period
Your grandmother has assets of $500,000. One option that she is considering involves an adult residential community for a six-year period beginning in 2009. The model
Describes yearly adult residential community costs n years after 2008. Does your grandmother have enough to pay for the facility?