# Mr. Germanis' Class Website

## Sequences and Series

 Date Entry Task Activity Assignments and Homework* 4/13 Work in your table groups to determine the next five numbers in the sequences. {4, 4, 8, 12, 20, 32, ... } {1, -2, 3, -4, 5, -6, ... } {0000, 0001, 0010, 0011, 0100, ...}. Additional Resources: Chapter 11.1 Stewart Pre-Calculus Khan Academy: Explicit and Recursive Definitions Intro to discrete mathematics. Sequences vs. Series Explicit formulas vs. recursion formula Sigma notation JR: Develop both recursive vs. explicit formula for the following sequence of numbers and explain your reasoning: 1, 4, 9, 16, 25, ... Notes from Today's Class Give the first four terms and the 31st term of the following sequences: an = 2n + 3 an = 1 + (-1)n Test Corrections: Tuesday (4/14)PM, Wednesday (4/15)AM. Today's Goals: Students will become familiar with the differences between sequences and series, learn about notation and both explicit/recursive formulas. 4/14 Determine the first five terms of the following sequence: a1 = 1 an = 3(an - 1 + 2) History of the Fibonacci Sequence Recursively Defined Sequences. Finding a Pattern activity. JR: Lucas numbers are developed similarly to Fibonacci numbers. Determine the first 10 numbers in the sequence. Complete Finding a Pattern activity. 1) Determine the first 10 terms of this sequence and graph your results (on paper): a1 = 1 a2 = 3 an = an - 1 - an - 2 2) Determine the nth term (formula) of this sequence (the first term given is when n = 1): Test Corrections: Today (4/14)PM, Tomorrow (4/15)AM Today's Goals: Students will recognize patterns of various sequences and will define them recursively. 4/15 Recall the sequence assigned for homework on 4/1 {1/0!, 1/1!, 1/2!, 1/3!, 1/4!, ...,}: Write a formula for the nth term of the sequence. Partial Sums & Sigma Notation. Using the Notation activity. The Harmonic Series Activity JR: Identify and explain in simple language each component of sigma notation. 1. Compute the sum of 2. Write the sum of the following using sigma notation: Test Corrections: Today AM Today's Goals: Students will practice using converting a sum to sigma notation and computing sums using notation for partial sums. Introduction to infinite sums. 4/17 Label each of the following as either a geometric or arithmetic sequence and explain your reasoning 1/9, 2/9, 3/9, 4/9, 5/9, ... 1/3, 2/9, 3/27, 4/81, 5/243, ... 1, 2, 4, 8, 16, ... 2, 4, 6, 8, 10, ... 2, 8, 18, 32, 50, ... Made In The Shade activity. JR: Explain in simple terms how to construct a sequence that represents the area of the shaded regions from today's activity. Solutions to Made In The Shade 1. When an object is allowed to fall freely near the surface of the earth, the gravitational pull is such that the object falls 16 feet in the first second, 48 feet in the next second, 80 feet in the next second and so on. Find the total distance a ball falls in 6 seconds Find the formula for the total distance a ball falls in n seconds. 2. A biologist is trying to find the optimal salt concentrations for the growth of a certain species of mollusk. She begins with a brine solution that has 4g/L of salt and increases the concentration by 10% every day. Let C0 denote the initial concentration and Cn the concentration after n days. Find the recursive definition of Cn. Find the salt concentration after 8 days. (Optional self check) How could you model this with an exponential equation and solve with logarithms? Today's Goals: Students will practice the geometric and arithmetic sequences. Additionally students will practice generating explicit formulas and recursive formulas from an application problem. 4/20 An architect designs a theater with 15 seats in the first row, 18 in the second row, 21 in the third row, etc.  If the theater must seat 870, how many rows must there be? Show all your work. Write the progression of seats in series notation. Explain how you arrived at your answer. Provide a sketch of the theater seating. Practice with Quiz Topics. JR: In the well-known song "The Twelve Days of Christmas," a person gives his sweetheart k gifts on the kth day for each of the 12 days of Christmas. The person also repeats each gift identically on each subsequent day. Thus, on the 12th day, the sweetheart receives a gift for the first day, 2 gifts for the second, 3 gifts for the third, and so on. Set-up a sum with Sigma notation that will model the total number of gifts received up to day k. Prepare for the Quiz tomorrow. 1) Write an explicit formula for the following sequence. {0, -2, 4, -6, 8, -10, 12, ...} 2) Write a recursive formula for the sequence above. 3) Use sigma notation to write the sum of the first 8 terms of the sequence above. 4) Calculate the sum from question 3. Today's Goals: Review challenging concepts and prepare for the unit assessment. 4/21 Get ready for the quiz!  Move to a seat where you have ample room, Obtain all the materials you need before class starts, Seat at most two at the square "cafe tables" Place the paper "blinders" between each pair of people. Put yourself in a positive mental state. Unit Quiz 9.1 Explain the similarities and differences between a sequence and a series. Calculate a sequence using both explicit formulas and recursive formulas. Use sigma notation to find a sum of elements in a sequence. Recognize patterns in sequences (or partial sums) to write a general formula. JR: Suppose you have 2 shirts, 3 pants and 4 jackets. An outfit has exactly one of each clothing type. Draw a diagram showing the number of different outfits you can assemble. Roll a (physical) pair of standard dice ten times and record the sum of the dice for each roll in your homework notebook.  Create an histogram of your results (sum versus frequency).  Sketch how you believe a histogram of the combined class' data will appear and explain the difference from your graph. Today's Goals: Assess student understanding of sequences and series. Transition into combination's unit.
*Unless otherwise noted, homework is due the next class day.

### Overarching Unit Goals:

• Understand the difference between a sequence and a series.
• Calculate a sequence using both explicit formulas and recursive formulas.
• Use sigma notation to find a sum of elements in a sequence.
• Recognize patterns in sequences (or partial sums) to write a general formula.