Describing Number and Geometric Patterns Objectives: Use inductive reasoning in continuing patterns Find the next term in an Arithmetic and Geometric sequence Vocabulary Inductive reasoning: make conclusions based on patterns you observe Conjecture: conclusion reached by inductive reasoning based on evidence Geometric Pattern: arrangement of geometric figures that repeat Arithmetic Sequence Formed by adding a fixed number to a previous term Geometric Sequence Formed by multiplying by a fixed number to a previous term

Geometric Patterns Arrangement of geometric figures that repeat Use inductive reasoning and make conjecture as to the next figure in a pattern Use inductive reasoning to find the next two figures in the pattern. Use inductive reasoning to find the next two figures in the pattern.

Do Now Geometric Patterns Describe the figure that goes in the missing boxes. Describe the next three figures in the pattern below.

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

Real-world and Geometric Sequence You drop a rubber ball from a height of 100 cm and it bounces back to lower and lower heights. Each curved path has 80% of the height of the previous path. Write a function rule to model the problem. Write a Function Rule A(n) = a· r n - 1 A(n) = 100 · .8 n - 1 What height will the ball reach at the top of the 5th path? A(n) = 100 · .8 n - 1 A(5) = 100 · .8 5 - 1 A(5) = 40.96 cm