ࡱ> ')&y bjbj 4:{{||D.<j||||WWWkmmmmmm$"WWWWW||W||kWk:,| kf6N W0R0#30#0#PWWWWWWWWWWWWWW0#WWWWWWWWW| : Arithmetic and Geometric Sequences A sequence is a list of numbers or objects, called terms, in a certain order. In an arithmetic sequence, the difference between one term and the next is always the same. This difference is called a common difference. The common difference is added to each term to get the next term. 2, 5, 8, 11, 14, This is an increasing arithmetic sequence with a common difference of 3. 32, 26, 20, 14, 8, This is a decreasing arithmetic sequence with a common difference of 6. If asked to extend either of these patterns (the next 3 terms), you need to know the last term and then you can go on from there. These are examples of recursive sequences. Example: What are the next three terms in the sequence? 1, 5, 9, 13, I can see that this is an arithmetic sequence with a common difference of 4. To get the next three terms, add 4 to 13 which equals 17, the next term in the sequence. Then add 4 to 17 to get the next term to get 21, etc. So the next three terms are 17, 21, and 25. In a recursive sequence, you need to know the previous term to get the next term. In an explicit sequence, you can calculate any term in a sequence in a direct way using the first term and the common difference. Use the following formula to find any term of an arithmetic sequence.  EMBED Equation.3  an = the term in the sequence you are trying to find (n represents the desired term number) a1 = the first term in the sequence d = the common difference Example: What is the 10th term of the following sequence? 1, 5, 9, 13,  EMBED Equation.3  So the 10th term of this sequence is 37. Example: What is the 12th term of the following sequence? 34, 31, 28, 25, 22,  EMBED Equation.3  The 12th term of this sequence is 1. Practice: Find the next three terms: 3, 10, 17, 24, 31, _____, _____, _____ Find the 25th term: 53, 50, 47, 44, 41, ________ Find the 20th term: 25, 40, 55, 70, 85, ________ Find the 75th term: 88, 81, 74, 67, 60, ________ A geometric sequence is a sequence of numbers where the ratio of consecutive terms is constant. This ratio is called the common ratio (r). Sometimes the terms of a geometric sequence get so large that you may need to express the terms in scientific notation rounded to the nearest tenth. 2, 6, 18, 54, This is an increasing geometric sequence with a common ratio of 3. 1,000, 200, 40, 8, This is a decreasing geometric sequence with a common ratio or 0.2 or  EMBED Equation.3 . Geometric sequences can also be recursive or explicit. Remember recursive means you need the previous term and the common ratio to get the next term. Example: What are the next three terms of the following sequence? 4, 20, 100, 500,  EMBED Equation.3 The next three terms are 2,500, 12,500, and 62,500. Explicit sequences also have a formula for finding any term in a sequence.  EMBED Equation.3  an = the term in the sequence you are trying to find (n represents the desired term number) a1 = the first term in the sequence r = the common ratio Example: Find the 7th term in the following sequence: 6, 18, 54, 162, Finding the common ratio can be harder than finding the common difference. One way to find it is the divide each term by the term before it.  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  So the common ratio is 3.  EMBED Equation.3  So the 7th term of the sequence is 4, 374. Example: Find the 8th term in the following sequence: 96, 48, 24, 12, 6, To find the common ratio, divide each term by the one before it.  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  The common ratio is  EMBED Equation.3 .  EMBED Equation.3  The 8th term of the sequence is 0.75. Practice: Find the next three terms: 128, 64, 32, 16, 8, _____, _____, _____ . Find the 9th term: 0.01, 0.1, 1, 10, 100, _________ Find the 7th term: 1, 6, 36, 216, 1,296, __________ Find the 11th term: 1, 2, 4, 8, 16, _________ Algebra Name ___________________ Arithmetic and Geometric Sequences Determine if the sequence is arithmetic or geometric, and then find the next three terms. 1. -2, -4, -8, -16, _____, _____, _____ 2. 65, 60, 55, 50, 45, _____, _____, _____ 3. 8, 13, 18, 23, 28, _____, _____, _____ 4. 1, 1.5, 2.25, 3.375, _____, _____, _____ Determine if the sequence is arithmetic or geometric, and then find the given term. 11th term: 5, 3, 1, 1, ____________ 23rd term: 0.1, 0.15, 0.2, 0.25, _____________ 6th term: 25, 75, 225, 675, ______________ 22nd term: 2, 5, 8, 11, 14, ____________ 10th term: a1 = 320, r = 0.5 50th term: 9, 2, 13, 24, 35, ______________ Mariano received a bonus of $50 for working the day after Thanksgiving, plus his regular wage of $9.45 an hour. If his total wages for the day were $135.05, how many hours did he work? 12. 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