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GEOMETRIC SEQUENCES ratio These are sequences where the ratio of successive terms of a sequence is always the same number. This number is called the common ratio.

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Geometric sequence A 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, , 81, 27, 9, 3, 1,

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A 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"

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Looking 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 term add 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

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Looking 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 term Many geometric sequences can me modeled with an exponential function an exponential function is a function of the form a n where a is the common ratio

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Here 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 2 1, 2 2, 2 3,... We can therefore model the sequence with the following formula: 2 n Check: When n = 1, which represents the first term, we get 2 1 = 2 When n = 2, which represents the second term, we get 2 2 = 2 × 2 = 4

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Let us try to model 243, 81, 27, 9, 3, 1, Common ratio? … divide by 3 Let n represent any term number in the sequence Observe that the terms of the sequence can be written as 3 5, 3 4, 3 3,... We just have to model the sequence: 5, 4, 3,.....

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Use arithmetic sequence to model 5, 4, 3, … – Common difference -1 The process will be briefly explained here The number we subtract to each term is 1 The number that comes right before 5 in the sequence is 6 We can therefore model the sequence with the following formula: -1* n , 81, 27, 9, 3, 1,

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We can therefore model 243, 81, 27, 9, 3, 1, with the exponential function below 3 -n + 6 Check: When n = 1, which represents the first term, we get = 3 5 = 243 When n = 2, which represents the second term, we get = 3 4 = , 81, 27, 9, 3, 1,

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Numerical Sequences and Patterns Arithmetic Sequence Add a fixed number to the previous term Find the common difference between the previous & next term Find the next 3 terms in the arithmetic sequence. 2,5,8,11,___,___,___ What is the common difference between the first and second term? Does the same difference hold for the next two terms?

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Arithmetic Sequence 17,13,9, 5, ___, ___, ___ What are the next 3 terms in the arithmetic sequence? An arithmetic sequence can be modeled using a function rule. What is the common difference of the terms in the preceding problem? -4 Let n = the term number Let A(n) = the value of the nth term in the sequence Term #1234n Term A(1) = 17 A(2) = 17 + (-4) A(3) = 17 + (-4) + (-4) A(4) = 17 + (-4) + (-4) + (-4) Relate Formula A(n) = 17 + (n – 1)(-4)

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Arithmetic Sequence Rule nth term first term number Common difference Find the first, fifth, and tenth term of the sequence: A(n) = 2 + (n - 1)(3) First Term A(1) = 2 + (1 - 1)(3) = 2 + (0)(3) = 2 A(n) = 2 + (n - 1)(3) Fifth Term A(5) = 2 + (5 - 1)(3) = 2 + (4)(3) = 14 A(n) = 2 + (n - 1)(3) Tenth Term A(10) = 2 + (10 - 1)(3) = 2 + (9)(3) = 29

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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)n Postage (cents) Real-world and Arithmetic Sequence What is the function rule? A(n) =.32 + (n – 1)(.23) What is the cost to mail a 10 ounce letter? A(10) =.32 + (10 – 1)(.23) =.32 + (9)(.23) = 2.39 The cost is $2.39.

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3,12,48,192,___, _____,______12,288 Numerical Sequences and Patterns Geometric Sequence Multiply by a fixed number to the previous term The fixed number is the common ratio Find the common ratio and the next 3 terms in the sequence. x x x 4 What is the common RATIO between the first and second term? Does the same RATIO hold for the next two terms?

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Geometric Sequence 80,20, 5,, ___, ___ What are the next 2 terms in the geometric sequence? 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 number Let A(n) = the value of the nth term in the sequence Term #1234n Term80205 A(1) = 80 A(2) = 80 · ( ¼ ) A(3) = 80 · (¼) · (¼) A(4) = 80 · (¼) · (¼) · (¼) Relate Formula A(n) = 80 · (¼) n-1

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Geometric Sequence Rule nth term first term common ratio Term number Find the first, fifth, and tenth term of the sequence: A(n) = 2 · 3 n - 1 First Term A(n) = 2 · 3 n - 1 Fifth Term A(n) = 2· 3 n - 1 Tenth Term A(1) = 2· A(5) = 2 · A(10) = 2· A(1) = 2A(5) = 162A(10) = 39,366

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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 personSegment = handshake Handshakes Segments …n…54321 Person Point (Term)

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What are Triangular Numbers? These are the first 100 triangular numbers:

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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=15...

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You can illustrate the name triangular number by the following drawing:

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1, 3, 6, 10, 15, 21, 28, 36, 45,... Triangular 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:

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We 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:

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The rectangles are n high and n+1 wide (and remember we doubled the dots), and x n is how many dots (the value of the Triangular Number n): 2x n = n(n+1) x n = n(n+1)/2 Rule: x n = n(n+1)/2

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Example: the 5th Triangular Number is x 5 = 5(5+1)/2 = 15 Example: the 60th is x 60 = 60(60+1)/2 = 1830 Wasn't it much easier to use the formula than to add up all those dots?

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