9-1 An Introduction to Sequences & Series

1. Draw a large triangle that takes up most of a full piece of paper. 2. Connect the (approximate) midpoints of all three sides to create a downward facing triangle. 3. With each remaining upward facing triangle repeat step Repeat step 3 keeping track of the number of upward facing triangles in each step.

Step# of triangles

The Rabbit Problem In the year 1202, a mathematician that will go unnamed for the time being, investigated the reproduction of rabbits. He created a set of ideal conditions under which rabbits could breed, and posed the question, "How many pairs of rabbits will there be a year from now?" The ideal set of conditions was a follows:

1. You begin with one male rabbit and one female rabbit. These rabbits have just been born. 2. A rabbit will reach sexual maturity after one month. 3. The gestation period of a rabbit is one month. 4. Once it has reached sexual maturity, a female rabbit will give birth every month. 5. A female rabbit will always give birth to one male rabbit and one female rabbit. 6. Rabbits never die

Let sub letter b represent baby (not sexually mature), and sub letter p represent pregnant Month # of Pairs Diagram 0 1

Rabbit Problem Solution Leonardo Fibonacci was the mathematician who posed the question. The conclusion came to be know as the Fibonacci sequence. So Or 466 rabbits

Fibonacci Spiral Orient your graph paper vertically Draw your first square on the 16th row up from the bottom and the 10 th column over. Continue squares in a counterclockwise order.

ViHart - Doodling in Math: MUkSXX0http:// MUkSXX0

What is a Sequence? A list of ordered numbers separated by commas.A list of ordered numbers separated by commas. Each number in the list is called a term.Each number in the list is called a term. For Example:For Example: Sequence 1 Sequence 2 2,4,6,8,10 2,4,6,8,10,… Term 1, 2, 3, 4, 5Term 1, 2, 3, 4, 5 Domain – relative position of each term (1,2,3,4,5) Usually begins with position 1 unless otherwise stated. Range – the actual terms of the sequence (2,4,6,8,10)

Sequence 1 Sequence 2 2,4,6,8,102,4,6,8,10,… 2,4,6,8,102,4,6,8,10,… A sequence can be finite or infinite. The sequence has a last term or final term. (such as seq. 1) The sequence continues without stopping. (such as seq. 2) Both sequences have a general rule: a n = 2n where n is the term # and a n is the nth term. The general rule can also be written in function notation: f(n) = 2n

Examples: Write the first 6 terms of a n =5-n.Write the first 6 terms of a n =5-n. a 1 =5-1=4a 1 =5-1=4 a 2 =5-2=3a 2 =5-2=3 a 3 =5-3=2a 3 =5-3=2 a 4 =5-4=1a 4 =5-4=1 a 5 =5-5=0a 5 =5-5=0 a 6 =5-6=-1a 6 =5-6=-1 4,3,2,1,0,-14,3,2,1,0,-1 Write the first 6 terms of a n =2 n.Write the first 6 terms of a n =2 n. a 1 =2 1 =2a 1 =2 1 =2 a 2 =2 2 =4a 2 =2 2 =4 a 3 =2 3 =8a 3 =2 3 =8 a 4 =2 4 =16a 4 =2 4 =16 a 5 =2 5 =32a 5 =2 5 =32 a 6 =2 6 =64a 6 =2 6 =64 2,4,8,16,32,642,4,8,16,32,64

Examples: Write a rule for the nth term. The seq. can be written as: Or, a n =2/(5 n ) The seq. can be written as:The seq. can be written as: 2(1)+1, 2(2)+1, 2(3)+1, 2(4)+1,… Or, a n =2n+1

Example: write a rule for the nth term. 2,6,12,20,…2,6,12,20,… Can be written as:Can be written as: 1(2), 2(3), 3(4), 4(5),… Or, a n =n(n+1)

Series A Series is The sum of the terms in a sequence.A Series is The sum of the terms in a sequence. Can be finite or infiniteCan be finite or infinite For Example:For Example: Finite Seq.Infinite Seq. 2,4,6,8,102,4,6,8,10,… Finite SeriesInfinite Series …

Summation Notation Also called sigma notationAlso called sigma notation (sigma is a Greek letter Σ meaning “sum”) The series can be written as: Sometimes you will see an n or k here instead of i. The notation is read: “the sum from i=1 to 5 of 2i” What if this sum was infinite? i goes from 1 to 5.

Summation Notation for an Infinite Series Summation notation for the infinite series:Summation notation for the infinite series: … would be written as: Because the series is infinite, you must use i from 1 to infinity (∞) instead of stopping at the 5 th term like before.

Examples: Write each series in summation notation. a …+100 Notice the series can be written as: 4(1)+4(2)+4(3)+…+4(25) Or 4(i) where i goes from 1 to 25. Notice the series can be written as:

Example: Find the sum of the series. (calc ok) k goes from 5 to 10.k goes from 5 to 10. (5 2 +1)+(6 2 +1)+(7 2 +1)+(8 2 +1)+(9 2 +1)+( )(5 2 +1)+(6 2 +1)+(7 2 +1)+(8 2 +1)+(9 2 +1)+( ) = = = 361

A Study from 1980 to 2003 determined that the increase in population of the united states could be modeled by If n represents the number of years past 1980, use the model to approximate the population in 2003, 2013, and 2023 (in millions).

Why do we care? Sequence Example The population of the Philippines has been growing approximately as follows: Series Example: What will be the volume of waste produced in future years in the Philippines? million60 million90 million

H Dub 9-1 Pg. 649 #3-24 (3n), 33-36all, odd 1 st hw problem – figure out which hw problems to do

Special Formulas (shortcuts!) C represents a constant

Special Formulas (shortcuts!) C represents a constant

Special Formulas (shortcuts!) C represents a constant

Story Time… When another famous mathematician was in first grade, his teacher asked the class to add up the numbers one through a hundred (1+2+3 etc., all the way up to 100). Write out the teacher’s request in summation notation, then find the answer (no calculators!) Try to figure out an efficient way! Save for 9-2

The Story of Little Gauss 1 to 100

Special Formulas (shortcuts!) C represents a constant

Special Formulas (shortcuts!) C represents a constant

Special Formulas (shortcuts!) C represents a constant

Special Formulas (shortcuts!)

Example: Find the sum. Use the 3 rd shortcut!Use the 3 rd shortcut!

Graphing a Sequence Think of a sequence as ordered pairs for graphing. (n, a n )Think of a sequence as ordered pairs for graphing. (n, a n ) For example: 3,6,9,12,15For example: 3,6,9,12,15 would be the ordered pairs (1,3), (2,6), (3,9), (4,12), (5,15) graphed like points in a scatter plot * Sometimes it helps to find the rule first when you are not given every term in a finite sequence. Term # Actual term