11-1 Mathematical Patterns Hubarth Algebra II. a. Start with a square with sides 1 unit long. On the right side, add on a square of the same size. Continue.

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11-1 Mathematical Patterns Hubarth Algebra II

a. Start with a square with sides 1 unit long. On the right side, add on a square of the same size. Continue adding one square at a time in this way. Draw the first four figures of the pattern. b. Write the number of 1-unit segments in each figure above as a sequence. c. Predict the next term of the sequence. Explain your choice. Each term is 3 more than the preceding term. The next term is , or 16. There will be 16 segments in the next figure in the pattern. 4, 7, 10, 13,... Ex. 1 Generating a Sequence

Ex. 2 Real-World Connection Suppose you drop a ball from a height of 100 cm. It bounces back to 80% of its previous height. How high will it go after its fifth bounce? The ball will rebound about 32.8 cm after the fifth bounce. Original height of ball: 100 cm After first bounce: 80% of 100 = 0.80(100) = 80 After 2nd bounce: 0.80(80) = 64 After 3rd bounce: 0.80(64) = 51.2 After 4th bounce: 0.80(51.2) = After 5th bounce: 0.80(40.96) =

Ex. 3 Using a Recursive Formula a. Describe the pattern that allows you to find the next term in the sequence 2, 6, 18, 54, 162,.... Write a recursive formula for the sequence. Multiply a term by 3 to find the next term. A recursive formula is a n = a n – 1 3, where a 1 = 2. b. Find the sixth and seventh terms in the sequence. Since a 5 = 162 a 6 = = 486 and a 7 = = 1458 c. Find the value of a 10 in the sequence. The term a 10 is the tenth term. a 10 = a 9 3 = (a 8 3) 3 = ((a 7 3) 3) 3 = ((1458 3) 3) 3 = 39,366

The spreadsheet shows the perimeters of regular pentagons with sides from 1 to 4 units long. The numbers in each row form a sequence. ABCDE 1a 1 a 2 a 3 a 4 2Length of Side Perimeter a. For each sequence, find the next term (a 5 ) and the twentieth term (a 20 ). In the sequence in row 2, each term is the same as its subscript. Therefore, a 5 = 5 and a 20 = 20. In the sequence in row 3, each term is 5 times its subscript. Therefore, a 5 = 5(5) = 25 and a 20 = 5(20) = 100. Sometimes you can find the value of a term of a sequence without knowing the preceding Term. Instead you can use the number of the term to calculate its value. A formula that expresses the nth term in terms of n is an explicit formula. Ex. 4 Real-World Connection b. Write an explicit formula for each sequence. The explicit formula for the sequence in row 2 is a n = n. The explicit formula for the sequence in row 3 is a n = 5n.

Practice 1. Describe the patterns formed. Find the next three terms. a. 27, 34, 41, 48, …….b. 243, 81, 27, 9, ……. Add 7; 55, 62, Describe the pattern that allows you to find the next term in the sequence 2, 4, 6, 8, 10, …. Write a recursive formula for the sequence and find the 11 th and 15 th terms 1, 4, 9, 16, 25, 36