Class Opener: Identify each sequence as geometric, arithmetic, or neither, find the next two terms, if arithmetic or geometric write the equation for the nth term in each. 45, 90, 180, 360… 5, 6, 8, 11, 15… 30, 35, 40, 45… 2, 1, 0.5, 0.25… 5. 25, 50, 75, 100…

Homework Check: Make sure to follow along and fix any mistakes you made on the homework last night.

Stop Light Activity You will have 25 minutes to finish up your stop light worksheets. Remember you are receive a grade on participation in class and an over score on the work you complete.

Quick Quiz Suppose a balloon is filled with 5000 𝑐𝑚 3 of helium. It then loses one fourth of its helium each day. Write the geometric sequence that shows the amount of helium in the balloon at the start of each day for five days. What is the common ratio of the sequence? How much helium will be left in the balloon at the start of the tenth day.

Stand Up and Stretch

Arithmetic Series Use the following Sequence: 1, 2, 3, 4, …….., 97, 98, 99, 100 to answer each of the following questions: Is it arithmetic, geometric, of neither? a) Add the first and last terms of the sequence and write down the answer. Then add the second and next-to-last terms. Continue adding till you get to the middle of the sequence. b) What patterns do you notice in your answers to part a? Use your answer to Question 2 to find the sum of the sequence?

Series: A series is the expression for the sum of the terms in a sequence. Finite sequences and series have terms that you can count individually from 1 a final whole number n. Infinite sequences and series go on forever. You indicate this with ellipsis points…

Write the Related Series Use the finite sequence 2, 11, 20, 29, 38, 47. Write the related series. Then evaluate the series.

Student Check: Write the related series for each finite sequence. Then evaluate the series. 0.3, 0.6, 0.9, 1.2, 1.5, 1.8, 2.1, 2.4, 2.7, 3.0 100, 125, 150, 175, 200, 225

Arithmetic Series: An arithmetic series is series whose terms form an arithmetic sequence. When a sequence has many terms, or when you know only the first and last terms of the sequence, you can use a formula to evaluate the related series quickly.

Sum of a Finite Arithmetic Series: The sum 𝑆 𝑛 of a finite arithmetic series: 𝑎 1 + 𝑎 2 + 𝑎 3 …+ 𝑎 𝑛 𝑆 𝑛 = 𝑛 2 ( 𝑎 1 + 𝑎 𝑛 ) Where 𝑎 1 is the first term, 𝑎 𝑛 is the nth term, and n is the number of terms.

Example: Each series has eight terms. Evaluate each related series: 5, 13, 21,…, 61 1, -1, -3,…, -13 -13, -14.5, -16…, 11.1

Summation Notation You can use the summation symbol, Ʃ to write a series. Then you can use limits to indicate how many terms you are adding. Limits are the least and greatest integral values of n.

Summation Notation Breakdown

Writing in Summation Notation Use summation notation to write the series: 3+5+6+9… for 33 terms

Student Check Use summation notation to write each series for the specified number of terms: 1+2+3+…; n = 6 3+8+13+18…; n = 9

Finding the Sum of a Series Use the series: 𝑛=1 3 (5𝑛+1) Find the number of terms in the series. Find the first and last terms of the series. Evaluate the series.

Student Check: For each sum, find the number of terms, the first term, and the last term. Then evaluate the series: 𝑛=1 10 (𝑛−3) 𝑛=1 4 ( 1 2 𝑛+1)