Superlesson
Project 4-1
Advanced Algebra
Chapter 4, Patterns and Structure in Algebra
Arithmetic sequences and series can be used to analyze patterns in nature as diverse as orbits of comets and diameters of trees.
Part A, Arithmetic Sequences
1. Like planets, "periodic" comets rotate in well-defined orbits about the Sun. The length of time it takes a comet to complete one rotation about the Sun is called the comet's period. Find data on the first 50 periodic comets.
a. Find the period of Comet Olbers (#13P). Periods are given in years.
b. Find the years of Comet Olbers's successive returns to the position it occupied at the time of its discovery, from the year of its discovery to the year of its next expected sighting.
c. How do you know that the above terms form an arithmetic sequence?
d. Historical records show that a bright object was visible in the night sky throughout the world in the year 1682. Might the object have been Comet Olbers? Explain.
Part B, Arithmetic Series
2. Each year a tree trunk increases its diameter through the addition of a "ring" of new material. If weather, nutrition, and other growing conditions do not change, ring widths will remain fairly constant. If conditions change, ring widths will vary from year to year.
The figure below shows a cross-section of a 5-year-old tree that has grown through the addition of rings of constant width.
a. Find the mean ring width of "Tree 1" for the 10 years 1985-1994. Round to the nearest hundredth.
b. Assume that the value you calculated in 2a above gives the tree's radius at the end of Year 1, as well as the width of each new annual ring.Write an arithmetic sequence expressing annual increases in the tree's radius. What is the common difference?
c. Find the first five partial sums of the series based on the above sequence. What do the sums represent?
d. Find the area of the trunk's cross-section after Year 1. Then find the cross-sectional area that is added to the trunk during each of the next four years. Write areas in terms of π. How do you know that the sequence is arithmetic?
e. Use sigma notation to express the trunk's cross-sectional area after five years. What is the area?