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Understanding Recursive and Explicit Formulas in Sequences

This lesson focuses on defining sequences using recursive and explicit formulas. We'll analyze how a sequence can start with a specific value and a consistent rule, such as A(n) = A(n-1) + d, to find terms within the sequence. By practicing with examples, including sequences like 20, 26, 32, and finding terms like the 5th and 10th using these formulas, students will strengthen their understanding of sequences. Assignments will reinforce these concepts through targeted exercises.

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Understanding Recursive and Explicit Formulas in Sequences

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  1. Arithmetic Recursive and Explicit formulas I can write explicit and recursive formulas given a sequence. Day 2

  2. An ordered list of numbers defined by a starting value (number) and a rule to find the general term. Recursive Formula: review A(1)= first term A(n-1)= Previous term A(n)= General term or nth term Given the following recursive formula, find the first 4 terms. 20, 26, 32, 38 A(1)= 20 A(n)= A(n-1) + 6 1st term2nd term3rd term 4th term A(1)= 20 A(n-1) + 6 A(n)= A(n-1) + 6 A(n)= A(2)= A(2-1) + 6 A(3)= A(3-1) + 6 A(2)= A(1) + 6 A(3)= A(2) + 6 A(2)= 20 + 6 A(3)= 26 + 6 A(2)= 26 A(3)= 32

  3. Explicit Formula: a function rule that relates each term of the sequence to the term number. A(n) = A(1) + (n-1)d Common difference 1st term nth term Term number Write an explicit formula given the following sequence and then find the 5th term. 20, 26, 32, … 44 Find it without the formula: 20, 26, 32, ___, ____, 38 Now, write and use the formula to find the 5th term: A(n) = A(1) + (n -1)d 5 20 6 n = A( ) = + ( -1) 5 5 20 A(1) = A( 5) = 20 + (4)6 6 d = A( 5) = 44

  4. Write an explicit formula for each recursive formula. A(1) = 19 A(n) = A(n-1) + 12 A(1) = 5 A(n) = A(n-1) - 3 A(n) = A(1) + (n-1)d A(n) = A(1) + (n-1)d A(n) = + (n-1) 19 A(n) = + (n-1) 5 (-3) 12 Find the 2nd, and 10th terms of the sequence on the left.. 10 19 19 2 12 10 A() = + ( - 1) 12 A() = + ( -1) 2 A( 10) = 19 + (9) 12 A(2) = 19 + (1)12 A( 10) = 19 + 108 A(2) = 19 + 12 A( 10) = 127 A(2) = 31

  5. Write a recursive formula for each explicit formula. A(n) = 32 + (n -1)12 A(n) = 32 + (n -1)12 A(1) = A(n) = A(n-1) 32 + 12 A(n) = 10 + (n -1)(- 4) A(n) = 10 + (n -1)(- 4) A(1) = A(n) = A(n-1) 10 Assignment: Page 279: 38-44 evens, 46-53, 66-67, 76,77, 80 - 4

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