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Arithmetic Sequences and Series An introduction………… 1, 4, 7, 10, 13 35 2, 4, 8, 16, 32 9, 1, 7, 15 6.2, 6.6, 7, 7.4 , 3, 6 12 9, 3, 1, 1/ 3 20 / 3 1, 1/ 4, 1/16, 1/ 64 85 / 64 , 2.5, 6.25 9.75 27.2 3 9 62 Arithmetic Sequences Geometric Sequences ADD To get next term MULTIPLY To get next term Arithmetic Series Sum of Terms Geometric Series Sum of Terms Find the next four terms of –9, -2, 5, … Arithmetic Sequence 2 9 5 2 7 7 is referred to as the common difference (d) Common Difference (d) – what we ADD to get next term Next four terms……12, 19, 26, 33 Find the next four terms of 0, 7, 14, … Arithmetic Sequence, d = 7 21, 28, 35, 42 Find the next four terms of x, 2x, 3x, … Arithmetic Sequence, d = x 4x, 5x, 6x, 7x Find the next four terms of 5k, -k, -7k, … Arithmetic Sequence, d = -6k -13k, -19k, -25k, -32k Vocabulary of Sequences (Universal) a1 First term an nth term n number of terms Sn sum of n terms d common difference nth term of arithmetic sequence an a1 n 1 d sum of n terms of arithmetic sequence Sn n a1 an 2 Given an arithmetic sequence with a15 38 and d 3, find a1. x a1 First term 38 an nth term 15 n number of terms NA Sn sum of n terms -3 d common difference an a1 n 1 d 38 x 15 1 3 X = 80 Find S63 of 19, 13, 7,... -19 a1 First term 353 ?? an nth term n number of terms 63 x Sn sum of n terms 6 d common difference an a1 n 1 d ?? 19 63 1 6 ?? 353 n a1 an 2 63 19 353 2 Sn S63 S63 10521 Try this one: Find a16 if a1 1.5 and d 0.5 1.5 a1 First term x 16 an nth term n number of terms NA Sn sum of n terms 0.5 d common difference an a1 n 1 d a16 1.5 16 1 0.5 a16 9 Find n if an 633, a1 9, and d 24 9 a1 First term 633 an nth term x n number of terms NA Sn sum of n terms 24 d common difference an a1 n 1 d 633 9 x 1 24 633 9 24x 24 X = 27 The sum of the first n terms of an infinite sequence is called the nth partial sum. Sn n (a1 an) 2 Example 6. Find the 150th partial sum of the arithmetic sequence, 5, 16, 27, 38, 49, … a1 5 d 11 c 5 11 6 an 11n 6 a150 11150 6 1644 S150 150 5 1644 75 1649 123,675 2 Example 7. An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 seats in the second row, 22 seats in the third row, and so on. How many seats are there in all 20 rows? d 1 c 20 1 19 an a1 n 1 d a20 20 19 1 39 20 S 20 20 39 10 59 590 2 Example 8. A small business sells $10,000 worth of sports memorabilia during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 19 years. Assuming that the goal is met, find the total sales during the first 20 years this business is in operation. a1 10,000 d 7500 c 10,000 7500 2500 an a1 n 1 d a20 10,000 19 7500 152,500 20 S20 10,000 152,500 10 162,500 1,625,000 2 So the total sales for the first 2o years is $1,625,000 Geometric Sequences and Series Start 04/21/2014 1, 4, 7, 10, 13 35 2, 4, 8, 16, 32 9, 1, 7, 15 6.2, 6.6, 7, 7.4 , 3, 6 12 9, 3, 1, 1/ 3 20 / 3 1, 1/ 4, 1/16, 1/ 64 85 / 64 , 2.5, 6.25 9.75 27.2 3 9 62 Arithmetic Sequences Geometric Sequences ADD To get next term MULTIPLY To get next term Arithmetic Series Sum of Terms Geometric Series Sum of Terms Vocabulary of Sequences (Universal) a1 First term an nth term n number of terms Sn sum of n terms r common ratio nth term of geometric sequence an a1r n1 a1 r n 1 sum of n terms of geometric sequence Sn r 1 Find the next three terms of 2, 3, 9/2, ___, ___, ___ 3 – 2 vs. 9/2 – 3… not arithmetic 3 9/2 3 1.5 geometric r 2 3 2 9 9 3 9 3 3 9 3 3 3 2, 3, , , , 2 2 2 2 2 2 2 2 2 2 9 27 81 243 2, 3, , , , 2 4 8 16 If a1 1 2 , r , find a9 . 2 3 a1 First term 1/2 an nth term x n number of terms 9 Sn sum of n terms NA r common ratio 2/3 an a1r n1 1 2 x 2 3 9 1 28 27 128 x 8 8 23 3 6561