Presentation on theme: "This number is called the common ratio."— Presentation transcript:
1This number is called the common ratio. GEOMETRIC SEQUENCESThese are sequences where the ratio of successive terms of a sequence is always the same number.This number is called the common ratio.
2Geometric sequenceA geometric sequence, in math, is a sequence of a set of numbers that follow a pattern. We call each number in the sequence a term.For examples, the following are sequences: 2, 4, 8, 16, 32, 64,243, 81, 27, 9, 3, 1,
3A geometric sequence is a sequence where each term is found by multiplying or dividing the same value from one term to the next. We call this value "common ratio"
4Looking at 2, 4, 8, 16, 32, 64, ......., carefully helps us to make the following observation: As you can see, each term is found by multiplying 2, a common ratio to the previous termadd 2 to the first term to get the second term,but we have to add 4 to the second term to get 8.This shows indeed that this sequence is not created by adding or subtracting a common term
5Looking at 243, 81, 27, 9, 3, 1, carefully helps us to make the following observation:This time, to find each term, we divide by 3, a common ratio, from the previous termMany geometric sequences can me modeled with an exponential functionan exponential function is a function of the form an where a is the common ratio
6Here is the trick! 2, 4, 8, 16, 32, 64,Let n represent any term number in the sequence Observe that the terms of the sequence can be written as 21, 22, 23, ...We can therefore model the sequence with the following formula: 2nCheck:When n = 1, which represents the first term, we get 21 = 2When n = 2, which represents the second term, we get22 = 2 × 2 = 4
7Let us try to model 243, 81, 27, 9, 3, 1,Common ratio? … divide by 3Let n represent any term number in the sequenceObserve that the terms of the sequence can be written as 35, 34, 33, ...We just have to model the sequence: 5, 4, 3, .....
8243, 81, 27, 9, 3, 1,Use arithmetic sequence to model 5, 4, 3, …Common difference -1The process will be briefly explained hereThe number we subtract to each term is 1The number that comes right before 5 in the sequence is 6We can therefore model the sequence with the following formula:-1* n + 6
9with the exponential function below 243, 81, 27, 9, 3, 1,We can therefore model 243, 81, 27, 9, 3, 1,with the exponential function below3-n + 6 Check: When n = 1, which represents the first term, we get = 35 = 243 When n = 2, which represents the second term, we get = 34 = 81
10Numerical Sequences and Patterns Arithmetic SequenceAdd a fixed number to the previous termFind the common difference between the previous & next termExampleFind the next 3 terms in the arithmetic sequence.2, 5, 8, 11, ___, ___, ___141721+3+3+3+3+3+3What is the common difference between the first and second term?Does the same difference hold for the next two terms?
11What are the next 3 terms in the arithmetic sequence? 17, 13, 9, 5, ___, ___, ___1-3-7An arithmetic sequence can be modeled using a function rule.What is the common difference of the terms in the preceding problem?-4Let n = the term numberLet A(n) = the value of the nth termin the sequenceA(1) = 17A(2) = 17 + (-4)A(3) = 17 + (-4) + (-4)A(4) = 17 + (-4) + (-4) + (-4)Term #1234nTerm171395RelateFormula A(n) = 17 + (n – 1)(-4)
13Real-world and Arithmetic Sequence In 1995, first class postage rates were raised to 32 cents for the first ounce and 23 cents for each additional ounce. Write a function rule to model the situation.Weight (oz)A(1)A(2)A(3)nPostage (cents)What is the function rule?A(n) = (n – 1)(.23)What is the cost to mail a 10 ounce letter?A(10) = (10 – 1)(.23)= (9)(.23)= 2.39The cost is $2.39.
14Numerical Sequences and Patterns Geometric SequenceMultiply by a fixed number to the previous termThe fixed number is the common ratioExampleFind the common ratio and the next 3 terms in the sequence.3, 12, 48, 192, ___, _____, ______768307212,288x 4x 4x 4x 4x 4x 4Does the same RATIO hold for the next two terms?What is the common RATIO between the first and second term?
15What are the next 2 terms in the geometric sequence? 80, 20, 5, , ___, ___An geometric sequence can be modeled using a function rule.What is the common ratio of the terms in the preceding problem?Let n = the term numberLet A(n) = the value of the nth termin the sequenceA(1) = 80A(2) = 80 · (¼)A(3) = 80 · (¼) · (¼)A(4) = 80 · (¼) · (¼) · (¼)Term #1234nTerm80205RelateFormula A(n) = 80 · (¼)n-1
16Geometric Sequence Rule A(n) = a rTermnumbernthtermfirsttermcommonratioFind the first, fifth, and tenth term of the sequence:A(n) = 2 · 3n - 1First TermFifth TermTenth TermA(n) = 2· 3n - 1A(n) = 2 · 3n - 1A(n) = 2· 3n - 1A(1) = 2·A(5) = 2 ·A(10) = 2·A(1) = 2A(5) = 162A(10) = 39,366
17Point = 1 person Segment = handshake Hand Shake Problem:If each member of this class shook hands with everyone else, how many handshakes were there altogether?People in class =Mathematical Model:Point = 1 person Segment = handshakePerson Point(Term)12345…n…25HandshakesSegments13610300
18What are Triangular Numbers What are Triangular Numbers? These are the first 100 triangular numbers:
191 1+2=3 (1+2)+3=6 (1+2+3)+4=10 (1+2+3+4)+5=15 ... The sequence of the triangular numbers comes from the natural numbers (and zero), if you always add the next number:1 1+2=3 (1+2)+3=6 (1+2+3)+4=10 ( )+5=
20You can illustrate the name triangular number by the following drawing:
21Triangular Number Sequence This is the Triangular Number Sequence:This sequence is generated from a pattern of dots which form a triangle.By adding another row of dots and counting all the dots we can find the next number of the sequence:1, 3, 6, 10, 15, 21, 28, 36, 45, ...
22We can make a "Rule" so we can calculate any triangular number. First, rearrange the dots (and give each pattern a number n), like this:Then double the number of dots, and form them into a rectangle:
23The rectangles are n high and n+1 wide (and remember we doubled the dots), and xn is how many dots (the value of the Triangular Number n):2xn = n(n+1)xn = n(n+1)/2Rule: xn = n(n+1)/2
24Example: the 5th Triangular Number is Example: the 60th isx60 = 60(60+1)/2 = 1830Wasn't it much easier to use the formula than to add up all those dots?