Unit 1 – Section 4 “Recursive and Explicit Formula” Part 2 Objectives covered The student will determine an arithmetic sequence and write an equation based on its pattern.

Arithmetic Sequences 34,40,46,… 62,57,52,… 1 2 ,1 1 8 ,1 3 4 ,... Arithmetic Sequence: A sequence in which each term is found by adding or subtracting a constant called the common difference “d” to the previous term. Common Difference: The constant term that is used to add or subtract from one term to the next. EX: 26,21,16,…. What is the value of d? What are the next four terms of the sequence? Directions: Given the following sequences below, find the value of “d” and use it to find and list the next four numbers. 34,40,46,… 62,57,52,… 1 2 ,1 1 8 ,1 3 4 ,...

Recursive Formula N 1 2 3 4 F(n) 5 8 11 14 N 1 2 3 4 F(n) 20 12 -4 Recursive Formula does 2 things: Gives us the first term in the sequence. Gives an equation that shows how to find each term based on the previous term. Recursive Formula Variables 𝑎 1 =𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚 𝑎 𝑛−1 =𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑡𝑒𝑟𝑚 𝑑=𝑐𝑜𝑚𝑚𝑜𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒/𝑝𝑎𝑡𝑡𝑒𝑟𝑛 Recursive Formula: 𝑎 1 =𝐴 𝑎 𝑛 = 𝑎 𝑛−1 +𝑑 Steps to write a Recursive Formula Determine the 𝑎 1 , and write 𝑎 1 =_____ Determine the pattern to find d and replace d in the equation 𝑎 𝑛 = 𝑎 𝑛−1 +𝑑 Directions: Write a recursive formula for the following arithmetic sequences. 𝑎 1 =𝐴 𝑎 𝑛 = 𝑎 𝑛−1 +𝑑 1. 2. 3. N 1 2 3 4 F(n) 5 8 11 14 N 1 2 3 4 F(n) 20 12 -4

Explicit Function N 1 2 3 4 F(n) 5 8 11 14 N 1 2 3 4 F(n) 20 12 -4 Explicit Formula: This formula allows us to simply plug in the number of the term (or position) we are interested in, and we will get the value of that term. Explicit Formula Variables 𝑎 1 =𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚 𝑛 =𝑡𝑒𝑟𝑚 𝑜𝑟 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑑=𝑐𝑜𝑚𝑚𝑜𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒/𝑝𝑎𝑡𝑡𝑒𝑟𝑛 𝑓 𝑛 = 𝑎 1 + 𝑛−1 𝑑 Steps to write an Explicit Function Determine the 𝑎 1 , Determine the pattern to find d and replace d in the equation (Note: You are only replacing 𝑎 1 and d). 3. Distribute and simplify Directions: Write an explicit formula for the following arithmetic sequences. 𝑓 𝑛 = 𝑎 1 + 𝑛−1 𝑑 1. 2. 3. N 1 2 3 4 F(n) 5 8 11 14 N 1 2 3 4 F(n) 20 12 -4

Closing Questions Directions: Respond to the following questions. Be prepared to discuss your answers with the class. 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. Determine the explicit formula. Row 1 2 3 4 Seats 59 63 67 71

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