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Unit 2 Lesson 19 Arithmetic Sequences Objectives I can decipher a pattern in a number sequence I can write explicit and recursive formulas of arithmetic sequences Warm Up: write the next 3 terms of the sequence 1, 2, 2, 4, 8, 32….. 1, 1, 2, 3, 5, 8, 13, 21…… 3, 5, 7, 9….. 25, 22, 19, 16…… A sequence is an arrangement of numbers or objects that follow a rule or a pattern. Each number in a sequence is called a term. The variable a is often used to stand for the terms in a sequence. The first term is 𝒂𝟏 , the second is 𝒂𝟐 , and so on. The nth term of the sequence is written 𝒂𝒏 , where n can be any positive integer. In an arithmetic sequence, each term is found by adding a fixed number, called the common difference (d), to the previous term. Are any of the warm ups examples of arithmetic sequences? If so, what is d? There are two ways to write sequence formulas—recursive and explicit. In the recursive process, finding a term depends on knowing a previous term, so you MUST define at least one previous term. The recursive definition looks like this 𝒂𝒏 = 𝒂𝒏−𝟏 + d Let’s Try….recursive equation Let our sequence be 2, 4, 6, 8…. 𝒂𝒏 = 𝒂𝒏−𝟏 + d The explicit definition looks like this 𝒂𝒏 = 𝒂𝟏 + (n – 1)d Let’s Try….explicit equation 28, 24, 20, 16…. Write the explicit equation for this arithmetic sequence Write the recursive equation too!! In a certain arithmetic sequence, each term is found by subtracting 4.5 from the previous term. If the first term in the sequence is 10, what is the 9th term in the sequence. Write the explicit formula to find 𝑎𝑛 You try………. 5, 9, 13, 17, 21,….. write the recursive equation write the explicit formula find the 15th term 10, 3, -4, -11, -18,….. write the recursive equation write the explicit formula find the 18th term Ami is training for a long distance race. She ran for 30 minutes per day on three days this week. Each week she will increase her daily running time by 5 minutes. By the 6th week, for how many minutes will she run each day that she trains? Name: _________________________ Unit 2 Lesson 19 Determine if the sequence is arithmetic. If it is, identify the common difference. 1. 5, 3, 1, -1, -3, ….. 2. 2, 2, 4, 6, 10, ….. 3. 3/2, 5/2, 7/2, 9/2, 11/2,… 4. 3, 2.4, 1.8, 1.2,….. Write a recursive and an explicit formula for each sequence. 5. 11, 15, 19, 23, ….. 6. 100, 88, 76, 64,….. 𝑎𝑛 = _________________ 𝑎𝑛 = _________________ 𝑎𝑛 = _________________ 𝑎𝑛 = _________________ 7. 𝑎1 = 14 8. d=6 𝑎1 = 0 d = 1/3 𝑎𝑛 = _________________ 𝑎𝑛 = _________________ 𝑎𝑛 = _________________ 𝑎𝑛 = _________________ Find the indicated term in the arithmetic sequence. 9. If find 𝑎10 𝑎1 = 3 and 𝑎2 = 5 10. If 𝑎1 = 3 and d = -3 find 𝑎4 11. Granny is knitting a 2 meters long scarf. On day 1 she finished with 18 cm. On every following day she adds 4 cm to the scarf. How many days will it take her to finish the scarf?