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

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, ....... 243, 81, 27, 9, 3, 1, ...............

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"

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

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 an where a is the common ratio

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 21, 22, 23, ... We can therefore model the sequence with the following formula: 2n Check: When n = 1, which represents the first term, we get 21 = 2 When n = 2, which represents the second term, we get 22 = 2 × 2 = 4

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 35, 34, 33, ... We just have to model the sequence: 5, 4, 3, .....

243, 81, 27, 9, 3, 1, ............... 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 + 6

with the exponential function below 243, 81, 27, 9, 3, 1, ............... 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 3-1 + 6= 35 = 243 When n = 2, which represents the second term, we get 3-2 + 6= 34 = 81

Numerical Sequences and Patterns Arithmetic Sequence Add a fixed number to the previous term Find the common difference between the previous & next term Example Find the next 3 terms in the arithmetic sequence. 2, 5, 8, 11, ___, ___, ___ 14 17 21 +3 +3 +3 +3 +3 +3 What is the common difference between the first and second term? Does the same difference hold for the next two terms?

What are the next 3 terms in the arithmetic sequence? 17, 13, 9, 5, ___, ___, ___ 1 -3 -7 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 A(1) = 17 A(2) = 17 + (-4) A(3) = 17 + (-4) + (-4) A(4) = 17 + (-4) + (-4) + (-4) Term # 1 2 3 4 n Term 17 13 9 5 Relate Formula A(n) = 17 + (n – 1)(-4)

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

Real-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) n Postage (cents) .32 + 23 .32+.23+.23 .32+.23+.23+.23 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.

Numerical Sequences and Patterns Geometric Sequence Multiply by a fixed number to the previous term The fixed number is the common ratio Example Find the common ratio and the next 3 terms in the sequence. 3, 12, 48, 192, ___, _____, ______ 768 3072 12,288 x 4 x 4 x 4 x 4 x 4 x 4 Does the same RATIO hold for the next two terms? What is the common RATIO between the first and second term?

What 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 number Let A(n) = the value of the nth term in the sequence A(1) = 80 A(2) = 80 · (¼) A(3) = 80 · (¼) · (¼) A(4) = 80 · (¼) · (¼) · (¼) Term # 1 2 3 4 n Term 80 20 5 Relate Formula A(n) = 80 · (¼)n-1

Geometric Sequence Rule A(n) = a r Term number nth term first term common ratio Find the first, fifth, and tenth term of the sequence: A(n) = 2 · 3n - 1 First Term Fifth Term Tenth Term A(n) = 2· 3n - 1 A(n) = 2 · 3n - 1 A(n) = 2· 3n - 1 A(1) = 2· 31 - 1 A(5) = 2 · 35 - 1 A(10) = 2· 310 - 1 A(1) = 2 A(5) = 162 A(10) = 39,366

Point = 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 = handshake Person Point (Term) 1 2 3 4 5 … n … 25 Handshakes Segments 1 3 6 10 300

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

1 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 (1+2+3+4)+5=15 ...

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

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

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:

The 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)/2 Rule: xn = n(n+1)/2

Example: the 5th Triangular Number is Example: the 60th is x60 = 60(60+1)/2 = 1830 Wasn't it much easier to use the formula than to add up all those dots?