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Answer Key Name: ____________________________________ Date: __________________ ARITHMETIC SERIES COMMON CORE ALGEBRA II A series is simply the sum of the terms of a sequence. The fundamental definition/notion of a series is below. THE DEFINITION OF A SERIES If the set a1 , a2 , a3 ,... represent the elements of a sequence then the series, Sn , is defined by: n Sn ai i 1 In truth, you have already worked extensively with series in previous lessons almost anytime you evaluated a summation problem. Exercise #1: Given the arithmetic sequence defined by a1 2 and an an 1 5 , then which of the following is 5 the value of S5 ai ? a2 a1 5 2 5 3 a3 a2 5 3 5 8 i 1 (1) 32 (3) 25 (2) 40 (4) 27 a4 a3 5 8 5 13 (2) a5 a4 5 13 5 18 S5 a1 a2 a3 a4 a5 40 The sums associated with arithmetic sequences, known as arithmetic series, have interesting properties, many applications and values that can be predicted with what is commonly known as rainbow addition. Exercise #2: Consider the arithmetic sequence defined by a1 3 and an an 1 2 . The series, based on the first eight terms of this sequence, is shown below. Terms have been paired off as shown. (a) What does each of the paired off sums equal? Each pair has a sum of 20. (b) Why does it make sense that this sum is constant? It makes sense that this sum is constant because as one term increases by 2, the other decreases by 2. (c) How many of these pairs are there? 3 5 7 9 11 13 15 17 (d) Using your answers to (a) and (c) find the value of the sum using a multiplicative process. The number of pairs will always be half the total number of elements, in this case 4 pairs. 4 20 80 (e) Generalize this now and create a formula for an arithmetic series sum based only on its first term, a1 , its last term, an , and the number of terms, n. In general, each pair will have the sum of the first and last terms, a1 an . The number of pairs will always be half the number of total elements, or n . So, our sum will be given by S n n a1 an . 2 2 COMMON CORE ALGEBRA II, UNIT #5 – SEQUENCES AND SERIES – LESSON #4 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015 SUM OF AN ARITHMETIC SERIES Given an arithmetic series with n terms, a1 , a2 , ..., an , then its sum is given by: Sn n a1 an 2 Exercise #3: Which of the following is the sum of the first 100 natural numbers? Show the process that leads to your choice. (1) 5,000 S100 1 2 3 100 (3) 10,000 (2) 5,100 100 1 100 50 101 5050 2 (4) 5,050 (4) Exercise #4: Find the sum of each arithmetic series described or shown below. (b) The first term is 8 , the common difference, d, is 6 and there are 20 terms (a) The sum of the sixteen terms given by: 10 6 2 46 50 . S16 a20 8 19 6 106 16 10 50 2 S 20 8 40 320 20 8 106 2 10 98 980 (c) The last term is a12 29 and the common difference, d, is 3 . (d) The sum 5 8 11 77 . 77 5 3 n 1 77 5 3n 3 29 a1 11 3 29 a1 33 a1 4 77 3n 2 3n 75 n 25 12 S12 4 29 2 S 25 S12 6 25 150 25 5 77 1025 2 Exercise #5: Kirk has set up a college savings account for his son, Maxwell. If Kirk deposits $100 per month in an account, increasing the amount he deposits by $10 per month each month, then how much will be in the account after 10 years? a120 100 119 10 1290 S120 120 100 1290 60 1390 2 $83, 400 COMMON CORE ALGEBRA II, UNIT #5 – SEQUENCES AND SERIES – LESSON #4 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015 Answer Key Name: ____________________________________ Date: __________________ ARITHMETIC SERIES COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. Which of the following represents the sum of 3 10 87 94 if the arithmetic series has 14 terms? (1) 1,358 (3) 679 (2) 658 (4) 1,276 S14 14 3 94 7 97 679 2 (3) 2. The sum of the first 50 natural numbers is (1) 1,275 (3) 1,250 (2) 1,875 (4) 950 S 50 1 2 3 49 50 50 1 50 25 51 1275 2 (1) 3. If the first and last terms of an arithmetic series are 5 and 27, respectively, and the series has a sum 192, then the number of terms in the series is (1) 18 (3) 14 (2) 11 (4) 12 192 n n 5 27 192 32 2 2 (4) 16n 192 n 12 4. Find the sum of each arithmetic series described or shown below. (a) The sum of the first 100 even, natural numbers. S100 2 4 6 200 100 2 200 50 202 2 10,100 (c) A series whose first two terms are 12 and 8 , respectively, and whose last term is 124. (b) The sum of multiples of five from 10 to 75, inclusive. 75 10 5 n 1 75 10 5n 5 75 5n 5 5n 70 n 14 S14 14 10 75 7 85 595 2 (d) A series of 20 terms whose last term is equal to 97 and whose common difference is five. From the first two terms we can see that the common difference is d 4 124 12 4 n 1 124 12 4n 4 124 4n 16 4n 140 n 35 terms S 35 35 12 124 1, 960 2 a20 a1 19d 97 a1 19 5 97 a1 95 a1 2 S 20 20 2 97 990 2 COMMON CORE ALGEBRA II, UNIT #5 – SEQUENCES AND SERIES – LESSON #4 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015 5. For an arithmetic series that sums to 1,485, it is known that the first term equals 6 and the last term equals 93. Algebraically determine the number of terms summed in this series. 1485 n n 6 93 1485 99 99n 2970 n 30 2 2 APPLICATIONS 6. Arlington High School recently installed a new black-box theatre for local productions. They only had room for 14 rows of seats, where the number of seats in each row constitutes an arithmetic sequence starting with eight seats and increasing by two seats per row thereafter. How many seats are in the new black-box theatre? Show the calculations that lead to your answer. a14 a1 13d 8 13 2 34 seats in last row S14 14 8 34 7 42 294 total seats 2 7. Simeon starts a retirement account where he will place $50 into the account on the first month and increasing his deposit by $5 per month each month after. If he saves this way for the next 20 years, how much will the account contain in principal? a240 a1 239d 50 239 5 $1245 in the last month S 240 240 50 1245 120 1295 $155, 400 2 8. The distance an object falls per second while only under the influence of gravity forms an arithmetic sequence with it falling 16 feet in the first second, 48 feet in the second, 80 feet in the third, etcetera. What is the total distance an object will fall in 10 seconds? Show the work that leads to your answer. This is an arithmetic sequence whose terms are separated by a common difference of 32. So, we can find the last or 10th term: 10 d i 1 i 10 16 304 5 320 1, 600 ft 2 d10 d1 9 32 16 9 32 304 9. A large grandfather clock strikes its bell once at 1:00, twice at 2:00, three times at 3:00, etcetera. What is the total number of times the bell will be struck in a day? Use an arithmetic series to help solve the problem and show how you arrived at your answer. We can start adding up the number of times the clock strikes: 1 2 3 12 1 2 3 12 1 2 3 12 12 1 12 6 13 78 2 78 78 156 bell strikes per day COMMON CORE ALGEBRA II, UNIT #5 – SEQUENCES AND SERIES – LESSON #4 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015

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