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Algebra II/Trig Honors Unit 7 Day 1: Define and Use Sequences and Series Objective: To recognize and write rules for number patterns Find the next two terms. Describe the pattern you continued in words. -5, -2, 1, 4, _________, _________ . 1, 4, 9, 16, _________, _________ . 3, 9, 27, 81, _________, _________ . Key Terms: Sequence - ______________________________________________________________ ________________________________________________________________________ o If the domain is not specified, _________________________________________. o The values in the range are called the ___________________________________. Domain: 1 2 3 4 . . . n The position of each term Range: a1 a2 a3 a4 . . . a n Terms of the sequence Finite sequence - _______________________________________________________ o Ex. 2, 4, 6, 8 Infinite sequence - ______________________________________________________ o Ex. 2, 4, 6, 8, … A sequence can be specified by an equation, or rule. Both sequences above can be described by the rule a n 2n or f n 2n . Example 1: Write terms of sequences Write the first six terms of each sequence. a. an 2n 5 b. f n 3 n 1 Writing Rules - If the terms of a sequence have a recognizable pattern, then you may be able to write a rule for the n-th term of the sequence. Example 2: Write rules for sequences Describe the pattern, write the next term, and write a rule for the n-th term of the sequences. a. -1, -8, -27, -64, . . . . . b. 0, 2, 6, 12, . . . . . Graphing Sequences – To graph a sequence, let the horizontal axis represent the position numbers (the domain) and the vertical axis represent the terms (the range). Example 3: Solve a multi-step problem You work in a grocery store and are stacking apples in the shape of a square pyramid with 7 layers. Write a rule for the number of apples in each layer. Then graph the sequence. (Let the top of the pyramid be the first layer). More Key Terms: Series - ________________________________________________________________ o Finite series: 2 + 4 + 6 + 8 o Infinite series: 2 + 4 + 6 + 8 + . . . . Summation Notation - ___________________________________________________ 4 Finite Series: 2 + 4 + 6 + 8 = 2i i 1 Infinite Series: 2 + 4 + 6 + 8 + . . . = 2i i 1 Read as “the sum of 2i for values of i from 1 to 4” o The index of the summation is i (can be any letter – most common: i, k, and n) o The lower limit of summation is 1 o The upper limit of summation is 4 for the finite series, for the infinite series Example 4: Write series using summation notation Write the series using summation notation. a. 25 + 50 + 75 + . . . + 250 b. 1 2 3 4 .... 2 3 4 5 Example 5: Find the sum of a series 3 k 8 Find the sum of the series 2 k 4 Special Formulas for Special Series (note: The index MUST start at one in order to use the forms.) n 1. Sum of n terms of a constant: c cn i 1 n 2. Sum of first n positive integers: i i 1 nn 1 2 n 3. Sum of the squares of first n positive integers: i i 1 2 nn 12n 1 6 *Rewrite example 5 and use the formula to find the sum. Example 6: Use a formula for a sum How many apples are in the stack in Example 3? The rule for the number in each layer was tn = n 2 . Class Practice: Write the first six terms of the sequence. 1. tn = n + 4 2. an = (-2) n-1 3. f ( n) = n n +1 4. For the sequence 3, 8, 15, 24, … determine the pattern and write the next term. Graph the first five terms and write a rule for the nth term. Write the series using summation notation. 5. 5+10 +15+... +100 6. 1 4 9 16 + + + +... 2 5 10 17 7. 6 + 36 + 216 +1296 +... 8. 5+ 6 + 7+... +12 Find the sum of the series. 5 9. 34 å8k 11. k=1 i=1 7 10. å1 6 å( k 2 -1) 12. k=3 ån n=1 HW: Page 438 #4-56 (M4), 58-62 Answers: 1. 5,6,7,8,9,10 2. 1,-2,4,-8,16,-32 3. 20 5. å 5k 6. k=1 ¥ i2 å i 2 +1 7. i=1 ¥ å 6i 8. i=1 1 2 3 4 5 6 4. Next term = 35, rule an = n ( n + 2) , , , , , 2 3 4 5 6 7 8 å( 4 + i) i=1 9. 120 10. 130 11. 34 12. 21