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April 30 th copyright2009merrydavidson Happy Birthday to: 4/25 Lauren Cooper
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9.1 Sequences & Series SEQUENCE: A list that is ordered so that it has a 1 st term, a 2 nd term, a 3 rd term and so on. example: 1, 5, 9, 13, 17, … a 1 = 1; a 2 = 5; a 3 = 9, etc. The nth term is denoted by: a n The domain of a sequence is the set of positive integers. The nth term is used to GENERALIZE about other terms.
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The three dots mean that this sequence is INFINITE. example: 1, 5, 9, 13, 17, … example: 2, -9, 28, -65, 126 This is a FINITE sequence.
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“Series” uses + signs. Arithmetic Sequence Arithmetic Series 3, 8, 13, 18, 23 3 + 8 + 13 + 18 + 23
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Given a “rule” for a sequence, find the 1 st 5 terms. EXAMPLE 1:
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Example 2: Write the first 4 terms of the sequence.
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Example 3. Write the first six terms of the sequence if
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Factorial Notation n! = n(n – 1)(n – 2)…1 Special case: 0! = 1 8 math/prb/4/enter = 40,320
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Factorial Notation n! = n(n – 1)(n – 2)…1 Special case: 0! = 1
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Summation Notation The Greek letter sigma, instructs you to add up the terms of the sequence. Example of sigma notation
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Example 4. Starting with an i value of 1 and ending with an i value of 5, write the series, then add. 2 + 5 + 10 + 17 + 26 = 60 Find the sum of:
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Example 5. Starting with an k value of 3 and ending with a k value of 6, write the expanded sum. Notice: k=3 to k =6 is 4 terms 10 + 17 + 26 + 37 = 90 Find the sum of:
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Example 6. -1 + 0 + 1 + 8 + 27 = 35 Find the sum of: Notice there are 5 terms here because you are starting at zero.
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A sequence uses comma’s A series uses + signs Summation notation uses sigma sign
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1, 5, 9, 13, 17, … The common difference is 4 When the difference between successive terms of a sequence is always the same number, the sequence is called arithmetic. In other words, the terms increase (or decrease) by adding a fixed quantity “d”.
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Is this sequence arithmetic? Example 7: 2, -4, 8, -16, 32… No because we are multiplying by -2 each time.
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Is this sequence arithmetic? Example 8: -5, 7, 19, 31,… yes because we are adding 12 each time.
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Is the sequence defined by S n = 3n + 5 arithmetic? Example 9: Let n = 1, n = 2, n = 3, etc to generate the sequence. 8, 11, 14, 17… yes because we are adding 3 each time. Notice that the common difference is the “slope” of the function. Therefore linear functions are arithmetic!
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Is the sequence defined by S n = 4 - n arithmetic? Example 10: Let n = 1, n = 2, n = 3, etc to generate the sequence. 3, 2, 1, 0, … yes because we are subtracting 1 each time. d = -1 a 1 = 3 Therefore linear functions are arithmetic!
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Formula for the nth term of Arithmetic Sequence: “a” is the first term and “d” is the common difference
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11) Write the nth term of the sequence 2, 7, 12, 17,….. Step 1: find the common difference 5 Step 2: write down the formula Step 3: fill in the formula with what you know
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12) Write the nth term of the sequence -12, -9, -6, ….. Step 1: find the common difference 3 Step 2: write down the formula Step 3: fill in the formula with what you know
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Use when you know the first term, number of terms and common difference Use when you know first term, last term, and number of terms Notice: Both formulas need first term and number of terms. SUMMATION FORMULAS:
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13) Find the sum of the first 12 terms of: Find “d”.. d = 4 Pick which formula you want to use. Plug and Chug.
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14) Find the indicated partial sum of: Find the 1 st term.. a 1 = -1 Pick which formula you want to use. Plug and Chug. Find the 2 nd term. a 2 = 1 Find “d”. d = 2 Find the last term. a 26 = 51
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