Knight’s Charge

Unit 1 Day 5 Tuesday1/27/15 1. For an arithmetic sequence, find 𝑎 20 𝑎 8 = 19 .

given 𝑎 2 = 1 and 2. Find the sum of the first 200 terms of the sequence: 1,5,9,13, 17,… Arithmetic Sequences and Series

           

Check Homework

6) 12, 19, 26, 33, 40 10) a+9, a+12, a+15, a+18, a+21 14) -7k, -13k, -19k, -25k, -31k 18) 10, 521 22) 27 26) 13/14 30) -332 31) 36, 42, 48 32) -4,-1,2,5 33) 2 , 10+2 2 2 , 20+ 2 3 , 10 34) 1, 1.75, 2.5, 3.25, 4 35) 77

Check Homework

      36) 140 37) 8 38) 16 41) 231 cans 42) NO, 100 because each of the first 100 even numbers is one more than each of the first odd numbers.

43) a- 31 b-496

Practice --- 30 minutes!

Practice --- 30 minutes!

Consider this:

A Chinese emperor of the Ming dynasty could not play chess. In return for lessons, the emperor promised his tutor any reward he wanted. Being a humble man, but needing to ask for something, he asked for only one grain of rice, doubled, for every square on the chess board. That is, one grain on the first square, two grains on the second square, four grains on the third square, etc.

 How many grains would be on the last square? (A chess board has 8 rows and 8 columns)  How many total grains were there be?

 Why did the emperor have the tutor beheaded?

Intro to Geometric Sequences/Series

Example: Write an explicit formula for the sequence {3, 6, 12, 24, 48, …}.  

Note: this sequence is geometric with a common ratio (r) of 2.

Make a table of values for the terms of the sequence. Then graph the table. What do you notice about the graph?

It’s EXPONENTIAL…… Can you write the equation of the function/sequence now?

Yes, the equation is 𝑦 = 3 ∙ 2 𝑥−1 … So the formula for the sequence is 𝑎 𝑛 = 3 ∙ 2 𝑛−1 Geometric Sequences

Example: Write an explicit formula for the sequence {3, 6, 12, 24, 48, …}.

 So how could we write the formula WITHOUT having to graph it?  In general, the explicit formula for an arithmetic sequence is given by 𝑎 𝑛 = 𝑎 1 ∙ 𝑟 𝑛−1 .

Arithmetic Sequences

Example: Fill in the chart for each geometric sequence shown.

SEQUENCE IMPLICIT FORMULA EXPLICIT FORMULA 20 th term

1 4 , 1 2 , 1, 2, 4, … 𝑎 𝑛 = 729 ∙ 1 3 𝑛 𝑎 𝑛 𝑎 1 = 1 = 3𝑎 𝑛−1 Geometric Sequences

Example: 354,294 is the ______th term of the sequence {2,6,18,…..} Geometric Sequences

Example: Construct a sequence that has THREE geometric means between 6 and 384.

{6, _____, _____, _____, 384, …} Geometric Sequences

Sum of a Finite Geometric Series

 The sum of a finite geometric sequence with common ratio r is 𝑆 𝑛 = 𝑎 1 1−𝑟 𝑛 1−𝑟 where 𝑟 ≠ 1 .

 Example: Find the sum of the first 10 terms of the sequence {1, 3, 9, 27, 81, …}.

Geometric Series

Example: Find the sum of the first 9 terms of a sequence whose first term is 1 and whose common ratio is 3. Geometric Series

EXAMPLE: Find the sum of the first 7 terms of a geometric sequence given 𝑎 1 = 2, 𝑎 7 = 1458, and 𝑟 = 3 .

Geometric Series

Sigma Notation

Write each of the following series in sigma notation. If possible, find the sum.

 {2, 6, 18, 54 … }  { 2000, 400, 80, 16, 3.2, 0.64

, …}  {1, 2, 4, 8, 16, 32, 64, 128, … } Sigma Notation

Example: Find the rule for the k th term of each sequence. Determine whether the sequence converges or diverges. If it converges, find what value it converges to.

Victory Lap

Homework

 p. 667-668 #6-20, 25-27 ALL  Have notebooks set up by tomorrow (with dividers)!

 Quiz Thursday!