Lesson 1.2 Exercises

01_ch01_pre-calculas11_wncp_tr.qxd 5/27/11 1:41 PM Page 6 Lesson 1.2 Exercises A 3. Use each arithmetic sequence to write the first 4 terms of an arithmetic series. a) 2, 4, 6, 8, . . . 2ⴙ4ⴙ6ⴙ8 c) 4, 0, -4, - 8, . . . 4ⴙ0ⴚ4ⴚ8 b) -2, 3, 8, 13, . . . ⴚ2 ⴙ 3 ⴙ 8 ⴙ 13 1 1 1 d) 2, 4, 0, - 4, Á 1 1 1 ⴙ ⴙ0ⴚ 2 4 4 4. Determine the sum of the given terms of each arithmetic series. a) 12 + 10 + 8 + 6 + 4 12 ⴙ 10 ⴙ 8 ⴙ 6 ⴙ 4 ⴝ 40 b) -2 - 4 - 6 - 8 - 10 ⴚ2 ⴚ 4 ⴚ 6 ⴚ 8 ⴚ 10 ⴝ ⴚ30 5. Determine the sum of the first 20 terms of each arithmetic series. a) 3 + 7 + 11 + 15 + Á Use: Sn ⴝ n[2t1 ⴙ d(n ⴚ 1)] 2 Substitute: n ⴝ 20, t1 ⴝ 3, d ⴝ 4 S20 ⴝ 20[2(3) ⴙ 4(20 ⴚ 1)] 2 S20 ⴝ 820 b) -21 - 15.5 - 10 - 4.5 - Á Use: Sn ⴝ n[2t1 ⴙ d(n ⴚ 1)] 2 Substitute: n ⴝ 20, t1 ⴝ ⴚ21, d ⴝ 5.5 S20 ⴝ 20[2( ⴚ 21) ⴙ 5.5(20 ⴚ 1)] 2 S20 ⴝ 625 B 6. For each arithmetic series, determine the indicated value. a) -4 - 11 - 18 - 25 - Á ; b) 1 + 3.5 + 6 + 8.5 + Á ; determine S28 determine S42 Use: Sn ⴝ n[2t1 ⴙ d(n ⴚ 1)] 2 Substitute: n ⴝ 28, t1 ⴝ ⴚ 4, d ⴝ ⴚ 7 S28 ⴝ 28[2( ⴚ 4) ⴚ 7(28 ⴚ 1)] 2 S28 ⴝ ⴚ2758 Use: Sn ⴝ n[2t1 ⴙ d(n ⴚ 1)] 2 Substitute: n ⴝ 42, t1 ⴝ 1, d ⴝ 2.5 S42 ⴝ 42[2(1) ⴙ 2 .5(42 ⴚ 1)] 2 S42 ⴝ 2194.5 01_ch01_pre-calculas11_wncp_tr.qxd 5/27/11 1:41 PM Page 7 7. Use the given data about each arithmetic series to determine the indicated value. a) S20 = - 850 and t20 = - 90; determine t1 Use: Sn ⴝ n(t1 ⴙ tn) 2 Substitute: Sn ⴝ ⴚ850, n ⴝ 20, tn ⴝ ⴚ90 ⴚ850 ⴝ 20(t1 ⴚ 90) 2 ⴚ850 ⴝ 10t1 ⴚ 900 50 ⴝ 10t1 t1 ⴝ 5 c) Sn = - 126, t1 = - 1, and tn = - 20; determine n Use: Sn ⴝ n(t1 ⴙ tn) 2 Substitute: Sn ⴝ ⴚ126, t1 ⴝ ⴚ1, tn ⴝ ⴚ20 ⴚ126 ⴝ n( ⴚ 1 ⴚ 20) 2 ⴚ252 ⴝ ⴚ21n n ⴝ 12 b) S15 = 322.5 and t1 = 4; determine d Use: Sn ⴝ n[2t1 ⴙ d(n ⴚ 1)] 2 Substitute: Sn ⴝ 322.5, n ⴝ 15, t1 ⴝ 4 322.5 ⴝ 322.5 43 35 d ⴝ ⴝ ⴝ ⴝ 15[2(4) ⴙ d(15 ⴚ 1)] 2 7.5(8 ⴙ 14d) 8 ⴙ 14d 14d 2.5 d) t1 = 1.5 and t20 = 58.5; determine S15 Use tn ⴝ t1 ⴙ d(n ⴚ 1) to determine d. Substitute: tn ⴝ 58.5, t1 ⴝ 1.5, n ⴝ 20 58.5 ⴝ 1.5 ⴙ d(20 ⴚ 1) 57 ⴝ 19d dⴝ3 Use: Sn ⴝ n[2t1 ⴙ d(n ⴚ 1)] 2 Substitute: n ⴝ 15, t1 ⴝ 1.5, d ⴝ 3 15[2(1.5) ⴙ 3(15 ⴚ 1)] 2 15(3 ⴙ 42) ⴝ 2 S15 ⴝ S15 S15 ⴝ 337.5 8. Two hundred seventy-six students went to a powwow. The first bus had 24 students. The numbers of students on the buses formed an arithmetic sequence. What additional information do you need to determine the number of buses? Explain your reasoning. I need to know the number of students on the last bus, then I can n(t1 ⴙ tn) . Or, I need to know the common difference, d, 2 n[2t1 ⴙ d(n ⴚ 1)] then I can use the rule Sn ⴝ . In each rule, I substitute 2 use the rule Sn ⴝ what I know, then solve for n. 01_ch01_pre-calculas11_wncp_tr.qxd 5/27/11 1:41 PM Page 8 9. Ryan’s grandparents loaned him the money to purchase a BMX bike. He agreed to repay $25 at the end of the first month, $30 at the end of the second month, $35 at the end of the third month, and so on. Ryan repaid the loan in 12 months. How much did the bike cost? How do you know your answer is correct? Ryan’s repayments form an arithmetic series with 12 terms, where the 1st term is his first payment, and the common difference is $5. n[2t1 ⴙ d(n ⴚ 1)] 2 12[2(25) ⴙ 5(12 ⴚ 1)] ⴝ 2 Use: Sn ⴝ S12 Substitute: n ⴝ 12, t1 ⴝ 25, d ⴝ 5 S12 ⴝ 6(50 ⴙ 55) S12 ⴝ 630 The bike cost $630. I used a calculator to add the 12 payments to check that the answer is the same. 10. Determine the sum of the indicated terms of each arithmetic series. a) 31 + 35 + 39 + Á + 107 Use tn ⴝ t1 ⴙ d(n ⴚ 1) to determine n. Substitute: tn ⴝ 107, t1 ⴝ 31, and d ⴝ 4 107 ⴝ 31 ⴙ 4(n ⴚ 1) 76 ⴝ 4n ⴚ 4 80 ⴝ 4n n ⴝ 20 Use: Sn ⴝ n(t1 ⴙ tn) 2 Substitute: n ⴝ 20, t1 ⴝ 31, and tn ⴝ 107 S20 ⴝ 20(31 ⴙ 107) 2 S20 ⴝ 1380 b) - 13 - 10 - 7 - Á + 62 Use tn ⴝ t1 ⴙ d(n ⴚ 1) to determine n. Substitute: tn ⴝ 62, t1 ⴝ ⴚ13, and d ⴝ 3 62 ⴝ ⴚ13 ⴙ 3(n ⴚ 1) 75 ⴝ 3n ⴚ 3 78 ⴝ 3n n ⴝ 26 Use: Sn ⴝ n(t1 ⴙ tn) 2 Substitute: n ⴝ 26, t1 ⴝ ⴚ13, and tn ⴝ 62 S26 ⴝ 26( ⴚ 13 ⴙ 62) 2 S26 ⴝ 637 11. a) Explain how this series could be arithmetic. 1 + 3 + ... This series could be arithmetic if each term was calculated by adding 2 to the preceding term. b) What information do you need to be certain that this is an arithmetic series? I need to know that the number added each time is 2. 01_ch01_pre-calculas11_wncp_tr.qxd 5/27/11 1:41 PM Page 9 12. An arithmetic series has S10 = 100, t1 = 1, and d = 2. How can you use this information to determine S11 without using a rule for the sum of an arithmetic series? What is S11? S11 ⴝ S10 ⴙ t11 I use tn ⴝ t1 ⴙ d(n ⴚ 1) to determine t11. I substitute n ⴝ 11, t1 ⴝ 1, d ⴝ 2. t11 ⴝ 1 ⴙ 2(11 ⴚ 1) t11 ⴝ 21 Then S11 ⴝ 100 ⴙ 21 S11 ⴝ 121 13. The side lengths of a quadrilateral form an arithmetic sequence. The perimeter is 74 cm. The longest side is 29 cm. What are the other side lengths? The sum of the side lengths form an arithmetic series with 4 terms, where t4 ⴝ 29 and S4 ⴝ 74. Use Sn ⴝ n(t1 ⴙ tn) to determine t1. 2 Substitute: Sn ⴝ 74, n ⴝ 4, tn ⴝ 29 74 ⴝ 4(t1 ⴙ 29) 2 74 ⴝ 2t1 ⴙ 58 16 ⴝ 2t1 t1 ⴝ 8 Use tn ⴝ t1 ⴙ d(n ⴚ 1) to determine d. Substitute: tn ⴝ 29, t1 ⴝ 8, n ⴝ 4 29 ⴝ 8 ⴙ d(4 ⴚ 1) 21 ⴝ 3d dⴝ7 So, the other side lengths are: 8 cm, 15 cm, and 22 cm 14. Derive a rule for the sum of the first n natural numbers: 1 + 2 + 3+ ... + n The sum of the numbers is an arithmetic series with t1 ⴝ 1, d ⴝ 1, and tn ⴝ n. n[2t1 ⴙ d(n ⴚ 1)] 2 n[2(1) ⴙ 1(n ⴚ 1)] Sn ⴝ 2 n(n ⴙ 1) Sn ⴝ 2 Use: Sn ⴝ Substitute: t1 ⴝ 1, d ⴝ 1 01_ch01_pre-calculas11_wncp_tr.qxd 5/27/11 1:41 PM Page 10 15. The sum of the first 5 terms of an arithmetic series is 170. The sum of the first 6 terms is 225. The common difference is 7. Determine the first 4 terms of the series. S5 ⴝ 170, S6 ⴝ 225 t6 ⴝ S6 ⴚ S5 ⴝ 225 ⴚ 170 ⴝ 55 Use tn ⴝ t1 ⴙ d(n ⴚ 1) to determine t1. Substitute: tn ⴝ 55, d ⴝ 7, n ⴝ 6 55 ⴝ t1 ⴙ 7(6 ⴚ 1) t1 ⴝ 20 So, t2 ⴝ 27, t3 ⴝ 34, and t4 ⴝ 41 The first 4 terms are: 20 ⴙ 27 ⴙ 34 ⴙ 41 C 16. The sum of the first n terms of an arithmetic series is: Sn = 3n2 - 8n Determine the first 4 terms of the series. Determine S1, S2, S3, and S4. In Sn ⴝ 3n2 ⴚ 8n: Substitute: n ⴝ 1 S1 ⴝ 3(1)2 ⴚ 8(1) ⴝ ⴚ5 Substitute: n ⴝ 3 S3 ⴝ 3(3)2 ⴚ 8(3) ⴝ3 t1 ⴝ S1 t2 ⴝ S2 ⴚ S1 ⴝ ⴚ5 ⴝ1 Substitute: n ⴝ 2 S2 ⴝ 3(2)2 ⴚ 8(2) ⴝ ⴚ4 Substitute: n ⴝ 4 S4 ⴝ 3(4)2 ⴚ 8(4) ⴝ 16 t3 ⴝ S 3 ⴚ S 2 t4 ⴝ S4 ⴚ S3 ⴝ7 ⴝ 13 17. Each number from 1 to 60 is written on one of 60 index cards. The cards are arranged in rows with equal lengths, and no cards are left over. The sum of the numbers in each row is 305. How many rows are there? Determine the sum of the first 60 natural numbers: 1 ⴙ 2 ⴙ 3 ⴙ . . . ⴙ 60 This is an arithmetic series with t1 ⴝ 1, d ⴝ 1, and t60 ⴝ 60. n(t1 ⴙ tn) 2 60(1 ⴙ 60) Sn ⴝ 2 Use: Sn ⴝ Substitute: n ⴝ 60, t1 ⴝ 1, tn ⴝ 60 Sn ⴝ 1830 The sum of the numbers in each row is 305, so the number of rows is: 1830 ⴝ6 305 01_ch01_pre-calculas11_wncp_tr.qxd 5/27/11 1:41 PM Page 11 18. Determine the arithmetic series that has these partial sums: S4 = 26, S5 = 40, and S6 = 57 t6 ⴝ S6 ⴚ S5 t5 ⴝ S 5 ⴚ S 4 t6 ⴝ 57 ⴚ 40 t5 ⴝ 40 ⴚ 26 ⴝ 17 ⴝ 14 t4 is: t5 ⴚ 3 ⴝ 11 t3 is: t4 ⴚ 3 ⴝ 8 t1 is: t2 ⴚ 3 ⴝ 2 The arithmetic series is: 2 ⴙ 5 ⴙ 8 ⴙ 11 d ⴝ t6 ⴚ t5 d ⴝ 17 ⴚ 14 ⴝ3 t2 is: t3 ⴚ 3 ⴝ 5 ⴙ 14 ⴙ 17 ⴙ . . .