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4­7 Sequences and Functions sequence (noun) SEE kwuns Definition: A sequence is an ordered list of numbers that often form a pattern. Each number in the list is called a term of the sequence. Example: The Fibonacci sequence is a sequence of numbers where the first number is 0, the second number is 1, and each subsequent number is equal to the sum of the previous two numbers. Origin: from the Latin word sequentia, which means “to follow” A sequence is an ordered list of numbers that often form a pattern. Each number is the list is called a term of a sequence. Essential Understanding When you can identify a pattern in a sequence, you can use it to extend the sequence. You can also model some sequences with a function rule that you can use to find any term of the sequence. In an arithmetic sequence, the difference between consecutive terms is constant. This difference is called the common difference. A sequence is a function whose domain is the natural numbers, and whose outputs are the terms of the sequence. You can write a sequence using a recursive formula. A recursive formula is a function rule that relates each term of a sequence after the first to the ones before it. consider the sequence 7, 11, 15, 19, …, you can use the common difference of the terms of an arithmetic sequence to write a recursive formula for the sequence. For the sequence 7, 11, 15, 19, …, the common difference is 4. Let n = the term number in the sequence. Let A (n) = the value of the n th term of the sequence. value of term 1 = A (1) = 7 Value of term 2 = A (2) = A (1) + 4 = 11 the common difference is 4. Value of term 3 = A (3) = A (2) + 4 = 15 Value of term 2 = A (4) = A (3) + 4 = 19 Value of term n = A (n) = A (n − 1) + 4 The recursive formula for the arithmetic sequence above is A (n) = A (n − 1) + 4 , where A (1) = 7 . You can find the value of any term of an arithmetic sequence using a recursive formula. You can also write a sequence using an explicit formula. An explicit formula is a function rule that relates each term of a sequence to the term number. Describe a pattern in each sequence. Then find the next two terms of the sequence. A.) 6, 13, 20, 27, . . . B.) 2, 6, 10, 14, . . . C.) 13, 11, 9, 7, . . . D.) 1.1, 2.2, 3.3, 4.4, . . . E.) 4.5, 9, 18, 36, . . . Tell whether the sequence is arithmetic. If it is, identify the common difference. F.) ­9, ­17, ­26, ­33, . . . G.) 2, 11, 21, 32, . . . H.) 0.2, 1.5, 2.8, 4.1, . . . I.) 10, 24, 36, 52, . . . J.) 15, 14.5, 14, 13.5, 13, . . . K.) ­3, ­7, ­10, ­14, . . . Write a recursive formula for each sequence. L.) 99, 88, 77, 66, . . . M.) 13, 10, 7, 4, . . . N.) 4.6, 4.7, 4.8, 4.9, . . . Write an explicit formula for each recursive formula. O.) A (n) = A (n − 1) + 3.4; A (1) = 7.3 P.) A (n) = A (n − 1) − 0.3; A (1) = 0.3 Write a recursive formula for each explicit formula. Q.) A (n) = 3 + (n − 1) (− 5) R.) A (n) = 4 + (n − 1) (1) Find the second, fourth, and eleventh terms of the sequence described by each explicit formula. S.) A (n) =− 3 + (n − 1) (5) T.) A (n) = 9 + (n − 1) (8) U.) A (n) =− 7 + (n − 1) (5) V.) A (n) =− 2.1 + (n − 1) (− 1.1) Tell whether each sequence is arithmetic. Justify your answer. If the sequence is arithmetic, write a recursive and an explicit formula to represent it. W.) ­3, ­7, ­11, ­15, . . . X.) ­5, 5, ­5, 5, . . . Y.) 0.2, ­0.6, ­1.4, ­2.2, . . . Using the recursive formula for each arithmetic sequence, find the second, third, and fourth term of the sequence. Then write the explicit formula that represents the sequence. Z.) A (n) = A (n − 1) + 1.2; A (1) = 8.8 AA.) A (n) = A (n − 1) − 2; A (1) = 0