Lesson 12.7: Sequences and Series May 30 7:11 AM Sequences Definition: A sequence is a set of numbers in a specific order. 2, 5, 8,. is an example of a sequence. Note: A sequence may have either a finite or an infinite number of terms. The terms of a sequence are the individual numbers in the sequence. If we let a 1 represent the first term of a sequence, a n represent the n th term, and n represent the term number, then the sequence is represented by a 1, a 2, a 3,, a n, In the example above, a 1 =2, a 2 =5, a 3 = 8, etc. Example: Write the first 3 terms of each sequence. 1. a n = n 2 + 2 2. a n = (n+1)(n+2) a 1 = 1 2 + 2 = 3 a 2 = 2 2 + 2 = 6 a 3 = 3 2 + 2 = 11 3. a n = 2a n 1 + 3, a 1 = 5 Mar 29 9:45 AM 1
arithmetic sequences Definition: An arithmetic sequence is a sequence in which each term, after the first, is found by adding a constant number to the previous term. That constant number is called the common difference. Example: In the sequence 2, 5, 8, 11, 14,... This is an arithmetic sequence because to get each number in the sequence, we add 3. So, we say the common difference is 3. Practice: Determine if the sequences below are arithmetic sequences. If yes, determine the common difference. 1.) 2, 3, 8,.. 2.) 5, 9, 13,.. 3.) 0, 7, 14, 4.) 5, 1, 7,. 5.) 9x, 2x, 5x,. Mar 29 9:50 AM How to write an arithmetic sequence: Example: Write the first five terms of the arithmetic sequence in which and d are given a 1 =17, d = 12 Start with a 1 and add the common difference (d) to the term. Continue adding to find more terms Practice: Write the first five terms of the arithmetic sequence in which a 1 and d are given as follows 1. a 1 = 19, d = 6 2. a 1 = 27, d = 4 3. a 1 = 3, d = Mar 29 11:44 AM 2
How to find any term of an arithmetic sequence If a 1 is the first term of an arithmetic sequence, a n the n th term, d is the common difference, a formula for finding the value of the n th term of an arithmetic sequence is: a n = a 1 + (n 1)d Example: Find the 75 th term of the sequence 2, 5, 8, 1. Find the 13 th term of 2, 8, 14, 20, 26,.. 2. Find the 43 rd term of 19, 15, 11,.. 3. Write the arithmetic sequence whose first term is 5 and whose th 7term is 17. 4. Find the common difference (d) in the arithmetic sequence whose st 1term is 4 and whose 11 th term is 64. 5. Find the common difference in the arithmetic sequence whose rd 3term is 0 and whose 29 th term is 50. Mar 29 11:49 AM Geometric Sequences Definition: A geometric sequence is a sequence in which each term, after the first is formed by multiplying the previous term by a constant number, which is called the common ratio. For example: 3, 6, 12, 24, is a geometric sequence, each term is being multiplied by 2 to produce the next term. a 1 represents the first term (3), a n the n th term, r is the common ratio (multiply by 2, so r = 2). To find r (the common ratio): divide any term in the sequence by the one before it. Example: Determine if the sequence is geometric. If it is, give the common ratio and the next two terms. 1. 2, 6, 18, 54, 162,. 2. 5, 10, 15, 20, 25,... 3. 10, 30, 90, 270, 810,... 4. 64, 32, 16, 8, 4,... Example: Write the first four terms of the geometric sequence in which a 1 and r are as follows. 5. = 13, r = 7 6. a 1 = 4, r = 7. =7, r = 2 Mar 29 12:10 PM 3
How to find any term of a geometric sequence If a 1 represents the first term of a geometric sequence, a n the n th term, and r the common ratio, then the formula for finding the n th term is: a n = a 1 r n 1 Example: 1) Find the 8 th term of the sequence 243, 81, 27, 9,. Practice: Find the term indicated in each of the following geometric sequence. 2. 10 th term of 3, 6, 12,.. 3. 6 th term of, 1, 2, 4, Mar 30 7:51 AM How to find r (the common ratio) Use the same formula: a n = a 1 r n 1 1. Find the common ratio of the geometric sequence, whose first term is 5 and whose 4 th term is 320. 2. Find the common ratio of the geometric sequence whose first term is 4 and whose 3 rd term is 36. Mar 30 7:58 AM 4
Series The sum of the terms of a sequence is called a series. Mar 28 5:39 PM Arithmetic Series To find the sum of a certain number of terms of an arithmetic sequence: S n = n(a 1 + a n ) 2 where S n is the sum of n terms (n th partial sum), a 1 is the first term, a n is the n th term. To find a n, you may have to use the formula for finding a term in an arithmetic sequence: a n = a 1 + (n 1)d Mar 28 5:44 PM 5
Example Find the sum of the first 20 terms of the sequence 4, 6, 8, 10,... Step 1: To use the sum formula, a n needs to be found first. a 1 = 4 n = 20 d = 2 ANSWER a n = a 1 + (n 1)d a 20 = 4 + (20 1)(2) a 20 = 4 + 19(2) a 20 = 4 + 38 a 20 = 42 Step 2: Now, we can use the sum formula to solve. S n = n(a 1 + a n ) 2 460 Mar 28 5:52 PM Practice 1. Find the sum of the first 30 terms of: 5, 9, 13, 17,... 2. Determine the sum of the first 17 terms of the arithmetic sequence whose first 4 terms are: 15, 9, 3, 3 3. Determine the sum of the first 8 terms of the arithmetic sequence whose first 4 terms are 8, 11, 14, 17 Mar 28 6:04 PM 6
4. Find the sum of the arithmetic series 3, 6, 9,...,99 Hint: Use the sum formula, n needs to be found first. 5. Determine the sum of 22, 16, 10,, 80 Mar 28 6:08 PM Geometric Series To find the sum of a certain number of terms of a geometric sequence: S n = a 1 (1 r n ) 1 r where S n is the sum of n terms (n th partial sum), a 1 is the first term, r is the common ratio. Mar 28 6:17 PM 7
Example Find the sum of the first 8 terms of the sequence 5, 15, 45, 135,... a 1 = 5 r = 3 n = 8 S n = a 1 (1 r n ) 1 r 8200 Mar 28 6:24 PM practıce 1) Determine the sum of the first 15 terms of the geometric sequence 1, 2, 4, 8, 2) Determine the sum of the first 11 terms of the geometric sequence 2, 6, 18, 54, 3) Determine the sum of the first 6 terms of the geometric sequence 1000, 200, 40, 8, 4) Determine the sum of the first 9 terms of the geometric sequence 1, 6, 36, 216, Mar 28 6:28 PM 8
Writing Series in Sigma Notation To write a series using sigma notation, look for patterns! See if the series is arithmetic or geometric. If it is arithmetic, use the form: here, d = common difference. Adjust c and r to fit the series you need. If it is geometric, use the form: here, r = common ratio. Again, adjust k and r accordingly. May 30 7:55 AM 1) Use sigma notation to represent the sum of the first 30 terms of the series: 4 + 8 + 12 + 16 +... 2) Use sigma notation to represent the sum of the first 15 terms of the series: 1 3 5 7 9... 3) Use sigma notation to represent the sum of the first 25 terms of the series: 2 1 + 2 2 + 2 3 + 2 4 +... 4) Use sigma notation to represent the sum of the first 40 terms of the series: 6 + 18 + 54 + 162 +... May 30 8:03 AM 9