Sequences and Series Checkpoint 1 Solutions

01_ch01_pre-calculas11_wncp_tr.qxd 5/27/11 1:41 PM Page 12 Checkpoint 1: Assess Your Understanding 1.1 1. Multiple Choice Which arithmetic sequence has d = - 8 and t10 = - 45? A. 27, 19, 11, 3, . . . B. -8, -12, -16, -20, . . . C. -5, -13, -21, -29, . . . D. -27, -19, -11, -3, . . . 2. Write the first 4 terms of an arithmetic sequence with its 5th term equal to -4. Sample response: I chose a common difference of 2. t5 ⴝ ⴚ4; so t4 is ⴚ4 ⴚ 2 ⴝ ⴚ6; t3 is ⴚ6 ⴚ 2 ⴝ ⴚ8; t2 is ⴚ8 ⴚ 2 ⴝ ⴚ10; and t1 is ⴚ10 ⴚ 2 ⴝ ⴚ12 My arithmetic sequence is: ⴚ12, ⴚ10, ⴚ8, ⴚ6, . . . 3. This sequence is arithmetic: -8, -11, -14, . . . a) Write a rule for the nth term. Use: tn ⴝ t1 ⴙ d(n ⴚ 1) tn ⴝ ⴚ8 ⴚ 3(n ⴚ 1) tn ⴝ ⴚ5 ⴚ 3n Substitute: t1 ⴝ ⴚ8, d ⴝ ⴚ3 b) Use your rule to determine the 17th term. For t17, use tn ⴝ ⴚ5 ⴚ 3n and substitute: n ⴝ 17 t17 ⴝ ⴚ5 ⴚ 3(17) t17 ⴝ ⴚ56 4. Use the given data about each arithmetic sequence to determine the indicated values. a) t4 = -5 and t7 = -20; determine d and t1 Use: t7 ⴝ t4 ⴙ 3d ⴚ 20 ⴝ ⴚ 5 ⴙ 3d ⴚ 15 ⴝ 3d d ⴝ ⴚ5 Use: t1 ⴝ t4 ⴚ 3d t1 ⴝ ⴚ5 ⴚ 3( ⴚ5) t1 ⴝ 10 Substitute: t7 ⴝ ⴚ20, t4 ⴝ ⴚ5 Substitute: t4 ⴝ ⴚ5, d ⴝ ⴚ5 Chapter 1: Sequences and Series—Checkpoint 1—Solutions 01_ch01_pre-calculas11_wncp_tr.qxd 5/27/11 1:41 PM Page 13 b) t1 = 3, d = 4, and tn = 59; determine n Use: tn ⴝ t1 ⴙ d(n ⴚ 1) 59 ⴝ 3 ⴙ 4(n ⴚ 1) 56 ⴝ 4n ⴚ 4 4n ⴝ 60 n ⴝ 15 Substitute: tn ⴝ 59, t1 ⴝ 3, d ⴝ 4 5. The steam clock in the Gastown district of Vancouver, B.C., displays the time on four faces and announces the quarter hours with a whistle chime that plays the tune Westminster Quarters. This sequence represents the number of tunes played from 1 to 3 days: 96, 192, 288, . . . Determine the number of tunes played in one year. In one year, there are 365 days and 96(365), or 35 040 quarters. So, in one year, 35 040 tunes are played. 1.2 6. Multiple Choice For which series could you use Sn = determine its sum? n(t1 + tn) to 2 A. 3 + 5 + 7 + 10 + 13 + 17 + 21 B. 3 - 1 - 5 - 9 - 13 - 17 - 21 C. -3 - 5 - 8 - 10 - 13 - 15 - 18 D. 3 - 1 + 5 - 3 + 7 - 5 + 9 7. a) Create the first 5 terms of an arithmetic series with a common difference of -3. Sample response: I chose a first term of 7. t1 ⴝ 7; so t2 is 7 ⴚ 3 ⴝ 4; t3 is 4 ⴚ 3 ⴝ 1; t4 is 1 ⴚ 3 ⴝ ⴚ2; and t5 is ⴚ2 ⴚ 3 ⴝ ⴚ5 My arithmetic series is: 7 ⴙ 4 ⴙ 1 ⴚ 2 ⴚ 5 ⴚ . . . b) Determine S26 for your series. Sample response: n[2t1 ⴙ d(n ⴚ 1)] 2 26[2(7) ⴚ 3(26 ⴚ 1)] ⴝ 2 Use: Sn ⴝ S26 Substitute: n ⴝ 26, t1 ⴝ 7, d ⴝ ⴚ3 S26 ⴝ ⴚ793 Chapter 1: Sequences and Series—Checkpoint 1—Solutions 01_ch01_pre-calculas11_wncp_tr.qxd 5/27/11 1:41 PM Page 14 8. Determine the sum of this arithmetic series: -2 + 3 + 8 + 13 + Á + 158 To determine n, use tn ⴝ t1 ⴙ d(n ⴚ 1) Substitute: tn ⴝ 158, t1 ⴝ ⴚ2, d ⴝ 5 158 ⴝ ⴚ2 ⴙ 5(n ⴚ 1) 160 ⴝ 5n ⴚ 5 165 ⴝ 5n n ⴝ 33 n(t1 ⴙ tn) Substitute: n ⴝ 33, t1 ⴝ ⴚ2, tn ⴝ 158 2 33( ⴚ 2 ⴙ 158) ⴝ 2 Use: Sn ⴝ S33 S33 ⴝ 2574 9. Use the given data about each arithmetic series to determine the indicated value. a) S17 = 106.25 and t17 = 8.25; determine t1 n(t1 ⴙ tn) Substitute: Sn ⴝ 106.25, n ⴝ 17, tn ⴝ 8.25 2 17(t1 ⴙ 8.25) 106.25 ⴝ 2 Use: Sn ⴝ 212.5 ⴝ 17t1 ⴙ 140.25 17t1 ⴝ 72.25 t1 ⴝ 4.25 b) S15 = 337.5 and t1 = -2; determine d Use: Sn ⴝ n(t1 ⴙ tn) to determine t15. 2 Substitute: Sn ⴝ 337.5, n ⴝ 15, t1 ⴝ ⴚ2 337.5 ⴝ 15( ⴚ 2 ⴙ t15 ) 2 675 ⴝ ⴚ30 ⴙ 15t15 705 ⴝ 15t15 t15 ⴝ 47 Use: tn ⴝ t1 ⴙ d(n ⴚ 1) 47 ⴝ ⴚ2 ⴙ d(15 ⴚ 1) 49 ⴝ 14d d ⴝ 3.5 Substitute: tn ⴝ 47, t1 ⴝ ⴚ2, n ⴝ 15 Chapter 1: Sequences and Series—Checkpoint 1—Solutions