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Advanced Algebra II Notes 1.1 Recursively Defined Sequences • Recursion: Process where each step of a pattern depends on the step before. • • Example A: • A square table seats 4 people. Two square tables pushed together seat 6 people. Three tables pushed together seat 8 people. How many people can sit at 10 tables arranged in a straight line? How many tables are needed to seat 32 people? Write a recursive definition to find the number of people who can sit at any linear arrangement of square tables. Arithmetic Sequence: a sequence in which each term is equal to the previous term plus a constant. This constant is called the common difference. If d is the common difference: u n u n1 d Table 1 4 2 6 Tables 3 4 8 10 5 12 6 14 7 8 16 18 9 10 20 22 Table 1 2 3 4 5 6 7 8 9 10 • Arithmetic Sequence: a sequence in which each term is equal to the previous term plus a constant. This constant is called the common difference. If d is the common difference: • • • • • • • • Sequence: Ordered list of numbers Term: each number in a sequence General term: nth term, un Recursive formula: Formula for a sequence which defines it in terms of relation to previous term. Example B • A concert hall has 59 seats in Row 1, 63 seats in Row 2, 67 seats in Row 3, and so on. The concert hall has 35 rows of seats. Write a recursive formula to find the number of seats in each row. How many seats are in Row 4? Which row has 95 seats? Row Seats 1 2 59 3 63 4 67 ... 71 ... u1 59 un un1 4 n2 Example C: The geometric pattern below is created recursively. If you continue the pattern endlessly, you create a fractal called the Sierpinski triangle. How many shaded triangles are there at stage 20? Stage Triangles 0 1 1 2 3 3 9 27 Geometric Sequence: a sequence in which each term is equal to the previous term multiplied by a constant. This constant is called the common ratio. If r is the ratio: un r un 1 u0 1 un 3 un1 n 1 Assignment: page 35: 1 - 12