Pre-Calc 11 Unit 1 Pre-Test

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1 Name: Class: Date: Pre-Calc 11 Unit 1 Pre-Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which term below is a term of an arithmetic sequence with t 9 = 156 and t 17 = 28? A. 11 C. 10 B. 13 D Determine the common difference, d, of this arithmetic sequence: 1,, 6, 16,... A. d = 1 C. d = 10 B. d = 10 D. The sequence has no common difference. 3. Determine the first terms of an arithmetic sequence, given the first term, t 1 =, and the common difference, d = 2. A., 6, 8, 10 C. 2, 2, 6, 10 B., 2, 0, 2 D. 2, 6, 10, 1. Two terms of an arithmetic sequence are t 6 = 25 and t 10 = 1. What is t 30? A. t 30 = 13 C. t 30 = 117 B. t 30 = 121 D. t 30 = Determine the common difference, d, of this arithmetic sequence:, 16, 36, 56,... A. d = C. d = 16 B. d = 20 D. d = In the arithmetic sequence: 18, 10, 2, 6,...; which term has the value 222? A. t 37 C. t 19 B. t 21 D. t Determine the number of terms in an arithmetic sequence with t 3 = 17.5 and t 10 = The final term in the sequence is A. n = 18 C. n = 13 B. n = 20 D. n = Determine the sum of the given terms of this arithmetic series: A. 10 C. 12 B. 520 D Determine the sum of the first 13 terms of this arithmetic series: A. S 13 = 13 C. S 13 = 2.5 B. S 13 = 52 D. S 13 = Determine the sum of the first 11 terms of an arithmetic series with t 1 = 5 and t 6 = 10. A. S 11 = 110 C. S 11 = B. S 11 = 57 D. S 11 =

2 Name: 11. An arithmetic series has S 21 = 2793, d = 13, and t 21 = 263; determine the first 3 terms of the series. A C B D An arithmetic series has t 1 = 5 and S 13 = 169; determine d. A. d = 3 C. d = 6 B. d = 2 D. d = An arithmetic series has t 1 = 7 and t 2 = 75; determine S 21. A. S 21 = 271 C. S 21 = 273 B. S 21 = 267 D. S 21 = The sum of the first 25 terms of an arithmetic series is The sum of the first 26 terms is The common difference is 5. Determine t 1. A. t 1 = 176 C. t 1 = 9 B. t 1 = D. t 1 = The sum of the first 15 terms of an arithmetic series is 210. The sum of the first 16 terms is The common difference is 2.. Determine the first terms of the series. A C B D Which sequence could be geometric? A. 6, 2, 2 3, 2 9,... C. 6, 2, 2 3, 2 9,... B. 6, 9 2, 18 5, 3,... D. 6, 3, 2, 3 2, is a term in which geometric sequence? A. 2,, 8, 16,... C. 2, 12, 72, 32,... B., 2, 1, 86,... D. 3, 6, 12, 2, In a geometric sequence, t 6 = 9375 and r = 5. Determine t 1. A. 1 C. 3 B. D Determine the 5th term of this geometric sequence: 108, 36, 12,,... 1 A. C. 3 9 B. D Determine r and t 6 of this geometric sequence: 32, 108, 36, 12,... A. r = 1 3, t 6 = 1 2 C. r = 3, t 6 = 27 B. r = 1 6, t 6 = 3 D. r = 1 3, t 6 = 3 2

3 Name: 21. In a finite geometric sequence, t 1 = 5 and t = 0. Determine t 10. A. 102 C B. 358 D In a finite geometric sequence, t 1 = 5 and t = 0. If the last term is 60, how many terms are in the sequence? A. 9 C. 10 B. 8 D To reproduce, a single bacterium splits into two. Assume it takes a single bacterium 20 min to split. When will there be 128 bacteria? A. 10 min C. 168 min B. 176 min D. 80 min 2. Jane deposited $1500 in a long-term savings account on her 23rd birthday. She did not make any more deposits or withdrawals. The account earned 7% per year. How much money did she have in the account when she turned 0? A. $28.25 C. $ B. $52.79 D. $ Determine the sum of the first 9 terms of this geometric series: Write the sum to the nearest hundredth if necessary. A. 7 C B. 76 D The sum of the first 9 terms of a geometric series is 513. The common ratio is 2. Determine the 1st term. A. 6 C. 3 B. 2 D Calculate the sum of this geometric series: A C B D Determine the number of terms in this geometric series: , then calculate its sum. A. n = 13; S 13 = 817 C. n = 11; S 11 = B. n = 12; S 12 = 8190 D. n = 1; S 1 = Determine the common ratio of a geometric series that has these partial sums: S = 3.5, S 5 = 3.75, S 6 = A. r = 1 C. r = 0.5 B. r = 2 D. r = The sum of the first 5 terms of a geometric series is 682. The common ratio is 1. Determine the 3rd term. A. 8 C. 128 B. 32 D

4 Name: 31. This infinite geometric series converges: Determine its sum. A. S = 21.3 B. S = 6 C. S = D. S = Determine whether this infinite geometric series has a finite sum: If it does, determine the sum. A. This series does not have a finite sum. C. S = 20.5 B. S = 15 D. S = An infinite geometric series has t 1 = 5 and r = 2. Determine S. A. S = 5 C. S = 2 B. The sum is not finite. D. S = 3. A geometric series has t 1 = 3 and S = 6. Determine the common ratio. A. r = 0.3 C. r = 0.5 B. r = 2 D. r = A geometric series has r = 3 and S = 16. Determine t 1. A. t 1 = C. t 1 = 16 B. t 1 = 12 D. t 1 = 36. Use an infinite geometric series to express 5. 5 as a fraction. A = 9 C = B = 55 D = The staircase to the Sky Pod level of Toronto s CN tower consists of 2579 steps. Suppose a person climbed all of the steps on the first day, and then each day after she climbed 7 of the steps she climbed on the 9 previous day until there were no more steps to climb. What is the total number of steps she climbed? Write your answer to the nearest step, if necessary. A. S = steps C. S = 9027 steps B. S = 3316 steps D. S = steps

5 Name: Short Answer 1. The sum of the first n terms of an arithmetic series is: S n = n 2 n Determine the first terms of the series. Problem 1. Tempel 1 is a periodic comet that orbits the Sun every 5.5 years and was last seen in the year If Tempel 1 maintains its current orbit around the Sun, will it appear in the year 2093? 2. The seating chart for a concert hall shows that a section of seats has 10 rows. Tickets in the first section sell for $95 each. Tickets in each consecutive section are $6 cheaper than the tickets in the preceding section. The concert hall has 100 rows. Joe wants to buy 1 ticket from each section to give to charity. How much money will he have to spend? 3. a) State the common ratio, then write the next 3 terms of this geometric sequence: 3, 6, 12, 2,... b) Identify the sequence as convergent or divergent. Explain. 5

6 Name:. The first term in a geometric sequence is Each term is double the previous term. What is term 25? 5. The sum of the first 6 terms of a geometric series is 315. The common ratio is 2. Determine t A robotic frog is programmed to make a sequence of jumps, each jump 1 the length of the preceding one. A frog is programmed to jump to a position 12 m away in 12 jumps. Determine the horizontal distance of each of the first 3 jumps. Give your answers to the nearest hundredth of a metre. 6

7 Pre-Calc 11 Unit 1 Pre-Test Answer Section MULTIPLE CHOICE 1. ANS: D PTS: 1 DIF: Moderate REF: 1.1 Arithmetic Sequences 2. ANS: B PTS: 1 DIF: Easy REF: 1.1 Arithmetic Sequences 3. ANS: A PTS: 1 DIF: Easy REF: 1.1 Arithmetic Sequences. ANS: B PTS: 1 DIF: Moderate REF: 1.1 Arithmetic Sequences 5. ANS: B PTS: 1 DIF: Easy REF: 1.1 Arithmetic Sequences 6. ANS: D PTS: 1 DIF: Moderate REF: 1.1 Arithmetic Sequences LOC: 11.RF9 TOP: Procedural Knowledge 7. ANS: D PTS: 1 DIF: Difficult REF: 1.1 Arithmetic Sequences 8. ANS: C PTS: 1 DIF: Easy REF: 1.2 Arithmetic Series 9. ANS: A PTS: 1 DIF: Easy REF: 1.2 Arithmetic Series 10. ANS: A PTS: 1 DIF: Easy REF: 1.2 Arithmetic Series 11. ANS: D PTS: 1 DIF: Moderate REF: 1.2 Arithmetic Series 12. ANS: A PTS: 1 DIF: Moderate REF: 1.2 Arithmetic Series 13. ANS: C PTS: 1 DIF: Moderate REF: 1.2 Arithmetic Series 1. ANS: B PTS: 1 DIF: Difficult REF: 1.2 Arithmetic Series 15. ANS: A PTS: 1 DIF: Difficult REF: 1.2 Arithmetic Series 16. ANS: C PTS: 1 DIF: Easy REF: 1.3 Geometric Sequences KEY: Procedural Knowledge 17. ANS: A PTS: 1 DIF: Difficult REF: 1.3 Geometric Sequences KEY: Procedural Knowledge 18. ANS: C PTS: 1 DIF: Moderate REF: 1.3 Geometric Sequences 19. ANS: D PTS: 1 DIF: Easy REF: 1.3 Geometric Sequences KEY: Procedural Knowledge 20. ANS: D PTS: 1 DIF: Moderate REF: 1.3 Geometric Sequences KEY: Procedural Knowledge 1

8 21. ANS: D PTS: 1 DIF: Moderate REF: 1.3 Geometric Sequences 22. ANS: B PTS: 1 DIF: Difficult REF: 1.3 Geometric Sequences 23. ANS: A PTS: 1 DIF: Difficult REF: 1.3 Geometric Sequences KEY: Problem-Solving Skills Procedural Knowledge 2. ANS: A PTS: 1 DIF: Difficult REF: 1.3 Geometric Sequences KEY: Problem-Solving Skills Procedural Knowledge 25. ANS: C PTS: 1 DIF: Moderate REF: 1. Geometric Series KEY: Procedural Knowledge 26. ANS: C PTS: 1 DIF: Easy REF: 1. Geometric Series KEY: Procedural Knowledge 27. ANS: B PTS: 1 DIF: Moderate REF: 1. Geometric Series 28. ANS: B PTS: 1 DIF: Moderate REF: 1. Geometric Series KEY: Procedural Knowledge 29. ANS: D PTS: 1 DIF: Moderate REF: 1. Geometric Series 30. ANS: B PTS: 1 DIF: Moderate REF: 1. Geometric Series KEY: Procedural Knowledge 31. ANS: A PTS: 1 DIF: Easy REF: 1.6 Infinite Geometric Series KEY: Procedural Knowledge 32. ANS: A PTS: 1 DIF: Easy REF: 1.6 Infinite Geometric Series 33. ANS: B PTS: 1 DIF: Easy REF: 1.6 Infinite Geometric Series KEY: Procedural Knowledge 3. ANS: C PTS: 1 DIF: Moderate REF: 1.6 Infinite Geometric Series KEY: Procedural Knowledge 35. ANS: D PTS: 1 DIF: Moderate REF: 1.6 Infinite Geometric Series KEY: Procedural Knowledge 36. ANS: C PTS: 1 DIF: Moderate REF: 1.6 Infinite Geometric Series KEY: Procedural Knowledge 37. ANS: A PTS: 1 DIF: Moderate REF: 1.6 Infinite Geometric Series KEY: Problem-Solving Skills Procedural Knowledge 2

9 SHORT ANSWER 1. ANS: PTS: 1 DIF: Difficult REF: 1.2 Arithmetic Series LOC: 11.RF9 TOP: Relations and Functions KEY: Conceptual Understanding PROBLEM 1. ANS: The years in which the comet appears form an arithmetic sequence. The arithmetic sequence for Tempel 1 has t 1 = 2005 and d = 5.5. To determine whether Tempel 1 should appear in 2093, determine whether 2093 is a term of its sequence. t n = t 1 + d(n 1) Substitute: t n = 2093, t 1 = 2005, d = = (n 1) Solve for n. 88 = 5.5n n = 93.5 n = 17 Since the year 2093 is the 17th term in the sequence, Tempel 1 should appear in PTS: 1 DIF: Moderate REF: 1.1 Arithmetic Sequences LOC: 11.RF9 TOP: Relations and Functions KEY: Communication Problem-Solving Skills 2. ANS: The prices of the tickets form an arithmetic sequence with first 3 terms $95, $89, $83,... The total cost of Joe s tickets is the sum of the first 10 terms of the arithmetic series: $95 + $89 + $ Use:S n = n[2t 1 + d(n 1)] 2 Substitute: n = 10, t 1 = 95, d = 6 10[2(95) 6(10 1)] S 10 = 2 S 10 = 680 To buy 1 ticket from each section, Joe will have to spend $680. PTS: 1 DIF: Difficult REF: 1.2 Arithmetic Series LOC: 11.RF9 TOP: Relations and Functions KEY: Communication Conceptual Understanding Problem-Solving Skills 3

10 3. ANS: a) r = 6 3,or 2 The next 3 terms are: t 5 = ( 2)( 2) = 8 t 6 = (8) ( 2) = 96 t 7 = ( 96)( 2) = 192 b) The sequence is divergent because the terms do not approach a constant value. PTS: 1 DIF: Moderate REF: 1.3 Geometric Sequences KEY: Communication Conceptual Understanding Problem-Solving Skills. ANS: The geometric sequence has first term and common ratio 2. Use:t n = t 1 r n 1 Substitute: t 1 = , r = 2, n = 25 t 25 = (2) 25 1 t 25 = Term 25 is PTS: 1 DIF: Moderate REF: 1.3 Geometric Sequences KEY: Communication Problem-Solving Skills 5. ANS: Use:S n = t (1 r n 1 ), r 1 Substitute: n = 6, S 1 r n = 315, r = 2 t Ê ˆ Ë Á 315 = = t 1 (1 6) = 63t 1 t 1 = 5 So, the first term is 5. PTS: 1 DIF: Moderate REF: 1. Geometric Series KEY: Problem-Solving Skills Procedural Knowledge

11 6. ANS: To determine t 1, use: S n = t (1 r n 1 ) 1 r Ê t Ë Á 12 = = 9 = t 1 12, r 1 Substitute: S 12 = 12, n = 12, r = 1 ˆ Ê 1 ˆ 1 Ë Á t 1 t 1 = The 1st jump is approximately 9.00 m. To determine t 2, use: t n = t 1 r n 1 Substitute: t 1 = , r = 1, n = 2 Ê t 2 = ˆ Ë Á 2 1 t 2 = The 2nd jump is approximately 2.25 m. To determine t 3, use: t n = t 1 r n 1 Substitute: t 1 = , r = 1, n = 3 Ê t 3 = ˆ Ë Á 3 1 t 3 = The 3rd jump is approximately 0.56 m. PTS: 1 DIF: Difficult REF: 1. Geometric Series KEY: Communication Conceptual Understanding Problem-Solving Skills 5