Sequences and Series

Transcription

1 Sequences and Series Arithmetic Sequences Find the common difference in sequence, and then write the next 3 terms in the sequence. 1) 3, 7,11, 15, 2) 1, 8, 15, 22, 3) 5, 2, -1, -4, 4) 68, 56, 44, 32, 5) 1.3, 2.6, 3.9, 5.2, Use the given information to write the first 5 terms of the sequence and the 20 th term. 6) a1= 4, d= 8 7) a1= 10, d= -6 8) a2= 9, d= 6 9) a15= 32, d= 5 10) 4, 7, 10, 13, Write the recursive formula for the sequence. 11) 3, 7,11, 15, 12) 5, 2, -1, -4, 13) 1.3, 2.6, 3.9, 5.2, 14) a2= 9, d= 6 15) a15= 32, d= 5 Write the explicit formula for the sequence. 16) a1= 4, d= 8 17) a1= 10, d= -6 18) a2= 9, d= 6 19) 1, 8, 15, 22, 20) 68, 56, 44, 32, Solve the following questions. 21) What is the common difference when a1=8 and a11=40? 22) What is 20 th term if a1=12 and a10= -33? 23) If a1=12, an=111,and d = 9, what is n? 24) Tom works at a car dealership selling cars. If he makes $4000 a month and $250 per car he sells, how much did he make in July if he sold 12 cars? In August he made $6500, how many cars did he sell? Find the missing term(s) in the sequence. 25) 3, _, 9 26) 4, _, _, 19 27) 5, _, _, -7 28) 20, _, _, _, 68 Homework Find the common difference in sequence, and then write the next 3 terms in the sequence. 29) 2, 11, 20, 29, 30) -17, -6, 5, 16, 31) 12, 9, 6, 3, 32) 45, 31, 17, 3, 33) 4.3, 5.7, 7.1, 8.5, Use the given information to write the first 5 terms of the sequence and the 20 th term. 34) a1= 4, d= 3 35) a1= 10, d= -8 36) a2= 9, d= 14 Alg II- Sequences & Series ~1~ NJCTL.org

2 37) a15= 32, d= 7 38) -1, 5, 11, 17, Write the recursive formula for the sequence. 39) a1= 4, d= 3 40) a1= 10, d= -8 41) 2, 11, 20, 29, 42) -17, -6, 5, 16, 43) 12, 9, 6, 3, Write the explicit formula for the sequence. 44) a15= 32, d= 7 45) -1, 5, 11, 17, 46) 12, 9, 6, 3, 47) 45, 31, 17, 3, 48) 4.3, 5.7, 7.1, 8.5, Solve the following questions. 49) What is the common difference when a1=3 and a11=44? 50) What is 20 th term if a1=2 and a10= -34? 51) If a1=22, an=78,and d = 8, what is n? 52) Tom works at a car dealership selling cars. If he makes $4000 a month and $250 per car he sells, how much did he make in July if he sold 16 cars? In August he made $10,000, how many cars did he sell? Find the missing term(s) in the sequence. 53) 4, _, 16 54) 3, _, _, 27 55) 12, _, _, -9 56) 12, _, _, _, 48 Geometric Sequences Find the common ratio in sequence, and then write the next 3 terms in the sequence. 57) 5, 10, 20, 40, 58) 4, -12, 36, -108, 59) 16, -8, 4, -2, 60) 6, 9, 13.5, 20.25, Use the given information to write the first 4 terms of the sequence and the 10 th term. 61) a1=6 and r=3 62) a1=12 and r=.5 63) a1=8 and r= 4 64) a3=20 and r=2 65) A cell reproduces by splitting in half every half hour, how many cells will there be in 6 hours if a the start there were 10? Write the recursive formula for the sequence. 66) a1=6 and r=3 67) a1=12 and r=.5 68) a1=8 and r= 4 69) 5, 10, 20, 40, 70) 4, -12, 36, -108, Write the explicit formula for the sequence. 71) 16, -8, 4, -2, 72) 6, 9, 13.5, 20.25, 73) a1=8 and r= 4 74) a3=20 and r=2 Alg II- Sequences & Series ~2~ NJCTL.org

3 Find the missing term(s) in the sequence. 75) 4, _, ) 5, _, _, ) 6, _, _, ) 7, _, _, _, 4375 Homework Find the common ratio in sequence, and then write the next 3 terms in the sequence. 79) 2, 8, 32, 128, 80) 6, -12, 24, -48, 81) 20, -10, 5, -2.5, 82) -4, 5, -6.25, , Use the given information to write the first 4 terms of the sequence and the 10 th term. 83) a1=8 and r=10 84) a1=20 and r=.5 85) a1=16 and r= -2 86) a3=36 and r=3 87) A cell reproduces by splitting in half every 15 minutes, how many cells will there be in 5 hours if a the start there were 20? Write the recursive formula for the sequence. 88) a1=8 and r=10 89) a1=20 and r=.5 90) 2, 8, 32, 128, 91) 6, -12, 24, -48, Write the explicit formula for the sequence. 92) a1=16 and r= -2 93) a3=36 and r=3 94) 20, -10, 5, -2.5, 95) -4, 5, -6.25, , Find the missing term(s) in the sequence. 96) 4, _, 36 97) 5, _, _, ) 6, _, _, ) 7, _, _, _, 1792 Geometric Series Find the indicated Sn. 100) a1=7, r=3, and n= 8, find S8 101) a1=6, r= -2, and n= 12, find S12 102) a1=8, r=.5, and n= 6 find S6 103) a1=4 and a8 = , find S8 104) a1=4 and a8 = , find S6 Evaluate the following 105) 4 n=1 2 3 n 1 106) 5 n=1 3 2 n 1 107) 6 n=1 2 ( 2) n 1 108) 6 (.5) n 1 5 n=1 Homework Find the indicated Sn. 109) a1=6, r=5, find S9 Alg II- Sequences & Series ~3~ NJCTL.org

4 110) a1=10, r= -4, find S7 111) a1=48, r=.5, find S8 112) a1=12 and a8 = 26244, find S8 113) a1=12 and a8 = 26244, find S6 Evaluate the following 114) 4 n=1 6 3 n 1 115) 6 n=1 8 (4) n 1 116) 7 n=1 10 (1.5) n 1 117) 12 ( 2) n 1 8 n=1 Special Sequences Write the first four terms of the sequence. 118) a1=11, a n = 2a n ) a1=-3, a2=2, a n = 4a n 1 3a n 2 Identify the sequence as arithmetic, geometric, or neither. Write the first 5 terms. 120) a1=3; an= 4an ) a1=6; an= -3an-1 122) a1=-4; an= an ) a1=-7; an= 2an ) a1=8; an= an ) a1=1; an= 6an-1 126) a1=-9;a2=6; an= 4an-1 + 2an-2 127) a1=10; a2= 8; an= -1an-1 + 2an-2 Home Work Write the first four terms of the sequence. 128) a1=-7; an= 2an ) a1=10; a2= 8; an= -1an-1 + 2an-2 Identify the sequence as arithmetic, geometric, or neither. Write the first 5 terms. 130) a1=2; an= an ) a1=8; an= -2an-1 132) a1=-14; an= an ) a1=-1; an= 6an-1 134) a1=12; an= 3an-1 135) a1=5; an=2 + 5an-1 136) a1=-4;a2=3; an= 2an-1 + 3an-2 Writing Sequences as Functions Write the sequence as a function and state the domain. 137) 20, -10, 5, -2.5, 138) -4, 5, -6.25, , 139) 12, 9, 6, 3, 140) 45, 31, 17, 3, Home Work 141) 2, 8, 32, 128, 142) 6, -12, 24, -48, 143) 16, -8, 4, -2, 144) 6, 9, 13.5, 20.25, Alg II- Sequences & Series ~4~ NJCTL.org

5 Review Multiple Choice 1) The sequence 2, 6, 10, 14, has a. a common ratio of 3 b. a common difference of 3 c. a common ratio of 4 d. a common difference of 4 2) The sequence 8, -4, 2, -1, has a. a common ratio of 1/2 b. a common difference of 1/2 c. a common ratio of -1/2 d. a common difference of -1/2 3) The next term in the sequence 7, 13, 19, 25 is a. 6 b. 31 c. 32 d. 48 4) The next term in the sequence 64, 48, 36, 27 is a. 20 b c d. 21 5) What is the common difference in this arithmetic sequence: 4,,,, 25 a b. 6.5 c. 7 d ) What is the common ratio in this geometric sequence: 4,,, 108 a. ±3 b. 3 c. ±27 d. 27 7) What is the common ratio in this geometric sequence: 32,,,, 2 a. ±.5 b..5 c. ±2 d. 2 8) a1 = 6 and d = 8 find S6 a. 46 b. 48 c. 108 d ) a1=8 and r = -4, find S6 a b c d ) n=4 3n = a. 24 b. 36 c. 72 d. 90 Alg II- Sequences & Series ~5~ NJCTL.org

6 11) 9 n=1 2n 5= a. 45 b. 40 c. 28 d ) 4 ( 1 n=1 2 )n 1 a. 0 b. 2.5 c. 2 2 /3 d. not possible 13) Find the first 5 terms of a1 = 3 and an= 2an-1 a. 3, 6, 9, 12, 15 b. 3, 5, 7, 9, 11 c. 3, 6, 12, 24, 36 d. 3, 6, 12, 24, 48 14) Find the first 5 terms of a1 = 2 and a2=8 and an= an-1 + an-2 a. 2, 8, 10, 18, 28 b. 2, 8, 10, 12, 20 c. 2, 8, 16, 32, 64 d. 2, 8, 16, 24, 32 15) The sequence 1, 1, 2, 3, 5, 8, is a. arithmetic b. geometric c. Fibonacci d. None of the above Extended Response 1) Your rich uncle wants to hire you to work for his company for 30 days. He offers to pay you (A) $1000 per day or (B) 1 cent the first day and double the pay each day their after. a. How much money do you make in 30 days of pay A? b. How much do you make on day 30 of pay B? c. How much money do you make in 30 days of pay B? 2) Given the sequence 2,,, -10 a. What are the missing terms if the sequence were geometric? b. What are the missing terms if the sequence were arithmetic? c. What is the sum of each sequence from part a and b. Use Sigma notation. 3) Given the arithmetic sequence of 4, 10, 16, 22, a. What is a20? b. What is an? c. What is S20? d. What is Sn? 4) Given the geometric sequence of 20, -10, 5, -2.5, a. What is a10? b. What is an? c. What is S20? d. What is Sn? e. What is S? Alg II- Sequences & Series ~6~ NJCTL.org

7 ANSWERS- SEQUENCES AND SERIES 1. D=4; 19, 23, =7; 29, 36, D=-3; -7, -10, D=-12; 20, 8, D=1.3; 6.5, 7.8, , 12, 20, 28, 36; , 4, -2, -8, -14; , 9, 15, 21, 27; , -33, -28, ; , 7, 10, 13, 16; an=an-1+4, a1=3 12. an=an-1-3, a1=5 13. an=an-1+1.3, a1= an=an-1+6, a1=9 15. an=an-1+5, a1= an=8n an=10n an=6n an=7n an=-12n D= ; 10 cars , , , 44, d=9; 38, 47, D=11; 27, 38, D=-3; 0, -3, D=-14; -11, -25, D=1.4; 9.9, 11.3, , 7, 10, 13, 16; , 2, -6, -14, -22; , 9, 23, 37, 51; , -59, -52, -45, -38; , 5, 11, 18, 25; an=an-1+3, a1=4 40. an=an-1-8, a1= an=an-1+9, a1=2 42. an=an-1+11, a1= an=an-1-3, a1= an=7n an=6n an=-3n an=14n an=1.4n dollars; 24 cars , , , 30, R=2; 80, 160, R=-3; 324, -972, R=-1/2; 1, -1/2, ¼ 60. R=1.5; ; , , 18, 54, 162; , 6, 3, 1.5; , 32, 128, 512; , 10, 20, 40; an=3an-1, a1=6 67. an=0.5an-1, a1= an=4an-1, a1=8 69. an=2an-1, a1=5 70. an=-3an-1, a1=4 71. an=16(-1/2) n an=6(1.5) n an=8(4) n an=5(2) n / , , /-35, 175, +/ R=4; 512, 2048, R=-2; 96, -192, R=-.5; 1.25, -.625, R=-1.25; , , 80, 800, 8000; 8,000,000, , 10, 5, 2.5; , -32, 64, -128; , 12, 36, 108; an=10an-1, a1=8 89. an=.5an-1, a1= an=4an-1, a1=2 91. an = -2an-1, a1=6 92. an=16(-2) n an=4(3) n an=20(-0.5) n an = -4(-1.25) n / , , /- 28, 112, +/ , , ,929, , , -18, 40, , 2, 17, N; 3, 14, 58, 234, G; 6, -18, 54, -162, A; -4, 4, 12, 20, N; -7, -15, -31, -63, A; 8, 10, 12, 14, G; 1, 6, 36, 216, N; -9, 6, 6, 36, N; 10, 8, 12, 4, , -15, -31, , 8, 12, A; 2, 10, 18, 26, G; 8, -16, 32, -64, A; -14, -19, -24, -27, G; -1, -6, -36, -216, G; 12, 36, 108, 324, N; 5, 27, 137, 687, N; -4, 3, -6, -3, = f(x)=20(-1/2) x-1, all real # s 138. f(x)=-4(-1/4) x-1, all real # s 139. f(x)=-3x+15, all real # s 140. f(x)=-4x+59, all real # s 141. f(x)=2(4) x-1, all real # s 142. f(x)=6(-2) x-1, all real # s 143. f(x)=16(-1/2) x-1, all real # s 144. f(x)=6(1.5) x-1, all real # s Alg II- Sequences & Series ~7~ NJCTL.org

8 Multiple Choice 1. D 2. C 3. B 4. B 5. A 6. B 7. A 8. D 9. C 10. D 11. A 12. C 13. D 14. A 15. C Extended Response 1. a b. 5,368,709 c. 10,737, a , b. -2, -6 c. 4 2 n=1 4 3 n 5 = n = 16 n=1 3. a. 118 b. an=6n-2 c d. S n = 17n + 3n 2 4. a b. a n = 20( 1 2 )n c. S 20 = d. S n = 13.33(1 ( 1 2 )n ) e. S = 40 3 Alg II- Sequences & Series ~8~ NJCTL.org