Name Honors Math 3 review material
October 2014 Chapter 2: Sequences and Series page 4
Review: Sequences and Series
Tomorrow we will have our end-of-unit test on sequences and series. For a topic list in textbook order, see page vii at the front of the textbook. For a list of the major things you learned how to do in each part of the chapter, see the bullet points on pp. 169-170. Here’s another outline of the material in which related skills and concepts are grouped together.
· Working with tables of numbers
o D columns for differences (sometimes repeated: D2, D3, etc.)
o S columns showing cumulative sums (sometimes repeated: SS, SSS, etc.)
· Sequence concept
o a sequence as a list of numbers
o a sequence as a function with domain of nonnegative integers (or all integers from some starting value)
o visual examples: figurate numbers
· Series concept
o definite and indefinite series
o writing in + notation or in S notation
o ways to visualize sums: staircases, pyramids
· Formulas for sequences and series
o closed-form formulas for the kth term of a sequence or for the sum of an indefinite series
o recursive formulas for the kth term of a sequence or for the sum of an indefinite series
· Evaluating definite and indefinite series
o arithmetic series using Gauss’s Method or its result
o geometric series using Euclid’s Method or its result
o Sk2, Sk3, Sk4, Sk5 using Bernoulli Formulas (these formulas will be given if needed)
o use of S identities to break series into simpler pieces that can be evaluated
· Limits
o limits of sequences
o limits of series
o limits of geometric series: formula
o visual examples: “fractals” (pp. 149-151 # 3, 4, 9)
o repeating decimals using limits of geometric series
· Pascal’s Triangle and Binomial Theorem
o patterns, relationships, and symmetry in Pascal’s Triangle
o summation properties: “hockey stick” (pp. 160-161), S/SS/SSS/etc. on diagonals (p. 163)
o Binomial Theorem: use and informal justification
Review and extension problems
11. Let rn represent the number of regions formed when n lines are drawn in a plane such that no two lines are parallel and no three lines are concurrent. The diagram below illustrates r1, r2,
and r3. Write a recursive definition for rn. Take it further: Also write a closed-form definition.
12. Let represent the number of intersection points created when n lines drawn in a plane such that no two lines are parallel and no three lines are concurrent. The diagrams above show that p1 = 0, p2 = 1, and p3 = 3.
a. Find p4 and p5.
b. Find a recursive definition for pn.
c. Find a closed-form definition for pn.
13. a. For what kind of problem would you be more likely to use a recursive definition of a sequence than a closed definition? Why?
b. For what kind of problem would you be more likely to use a closed definition of a sequence? Why?
14. a. A 3-by-3-by-3 cube shown has 27 little cubes but also many other cubes of various sizes. How many cubes does it have in all?
b. Find how many cubes of various sizes there would be in an
n-by-n-by-n cube.
15. A stack of oranges is completely arranged so the bottom layer consists of oranges in an equilateral triangle with n oranges on a side. The next layer to the bottom consists of an equilateral triangle of (n–1) oranges on a side. This pattern continues upward with one orange on the top. How many oranges are there?
16. Geometrically-formed limits such as the Koch curve from p. 149 #3 are known as fractals. Here’s another example of a fractal.
This figure is generated similarly to the Koch curve but with a different rule for how segments are replaced. The segment of Stage 0 gets replaced by 5 segments, each 1/3 the length of the original segment and arranged as shown in Stage 1. The same rule applies to each segment of Stage n as you pass to Stage n + 1.
a. Draw Stage 3.
b. Assume you start with length 1 at Stage 0. Find the length of the curve at Stages 1-3.
c. Find a closed form formula for the length of the curve at Stage n.
d. Does the length have a limit? Explain.
17. Optional extra: View these Vi Hart (youtube.com/user/vihart or vihart.com) videos where she draws fractals. Watch for things you learned about during this unit, plus much more.
http://www.youtube.com/watch?v=EdyociU35u8
http://www.youtube.com/watch?v=dsvLLKQCxeA
Tip from student K.C. — thanks!
Answers
These are newly-written answers; if you see anything that doesn’t look right, ask your teacher.
11. Each new line gives an additional number of regions equal to the total number of lines drawn. So r1 = 2 and rn = rn–1 + n for n > 1.
12. a. No two lines are parallel, so each new line intersects all previous lines. Thus
p4 = 3 + 3 = 6 and p5 = 6 + 4 = 10.
b. pn = pn–1 + n – 1.
c. Generally pn = the (n – 1)st triangular number = .
13. a. You would be more likely to use a recursive definition when you need to find the term following a given list of terms in a sequence.
b. You would be more likely to use a closed form definition when you need to find a term where n is very large.
14. a. A 3-by-3-by-3 cube contains 1 3-by-3-by-3 cube, 8 2-by-2-by-2 cubes, and 27 1-by-1-by-1 cubes or 36 cubes in all
b. An n-by-n-by-n cube contains 1 n-by-n-by n, 8 cubes of dimensions n-1, and 27 cubes of dimensions n-2. and so on, with 1-by-1-by-1 cubes;
15. Let f(k) stand for the number of oranges in the layer that has k oranges on the side.
For example, the picture shows that f(4) = 10. Generally, f(k) = .
Then to get the total number of oranges in the pyramid, add the layers:
= .
16. a. b. c.
d. No; as n increases, the lengths gets larger and larger without bound because they are given by a geometric sequence whose ratio is greater than 1.