#1-12: Write the first 4 terms of the sequence. (Assume n begins with 1.)

Transcription

1 Section 9.1: Sequences #1-12: Write the first 4 terms of the sequence. (Assume n begins with 1.) 1) a n = 3n a 1 = 3*1 = 3 a 2 = 3*2 = 6 a 3 = 3*3 = 9 a 4 = 3*4 = 12 3) a n = 3n 5 Answer: 3,6,9,12 a 1 = 3*1-5 = -2 a 2 = 3*2 5 = 1 a 3 = 3*3-5 = 4 a 4 = 3*4 5 = 7 Answer: -2,1,4,7 5) a n = 3 n a 1 = 3 1 = 3 a 2 = 3 2 = 9 a 3 = 3 3 = 27 a 4 = 3 4 = 81 Answer: 3,9,27,81 7) a n = (-3) n a 1 = (-3) 1 = -3 a 2 = (-3) 2 = 9 a 3 = (-3) 3 = -27 a 4 = (-3) 4 = 81 Answer: -3,9,-27,81 9) a n = (-1) n a 1 = (-1) 1 = -1 a 2 = (-1) 2 = 1 a 3 = (-1) 3 = -1 a 4 = (-1) 4 = 1 Answer: -1,1,-1,1

2 Section 9.1: Sequences 11) a n = a 1 = a 2 = a 3 = a 4 = Answer: #13-24: Find the indicated term 13) a n = 3n; a 8 Answer: a 8 = 3*8 = 24 15) a n = 3n 5; a 16 Answer: a 16 = 3*16-5 = 43 17) a n = 3 n ; a 5 Answer: a 5 = 3 5 = ) a n = (-3) n ; a 6 Answer: a 6 = (-3) 6 = ) a n = (-1) n ; a 50 Answer: a 50 = (-1) 50 = 1 23) a n = ; a 12 Answer: #25-40: write an expression for the n th term of the sequence 25) 2,4,6,8,10. Answer: a n = 2n

3 Section 9.1: Sequences 27) 3,5,7,9,11,13. I know I need a 2n since each term is 2 apart. Solve 2(1) x = 3 to figure out the number that goes after the 2n. Answer: a n = 2n+1 29) -3, -1, 1, 3,5 Each term is 2 apart, so again I need a 2n Solve 2(1) x = -3 to figure out what to write after the 2n Answer: a n = 2n ) 6,10,14,18,22 Each term is 4 apart, so I need a 4n. Solve 4(1) + x = 6 to figure out what number goes after the 4n. Answer: a n = 4n ) 7,13,19,25 Each term is 6 apart, so I need a 6n. Solve 6(1) + x = 7 to figure out what number goes after the 6n. Answer: a n = 6n +1 35) 2,4,8,16,32 I have to multiply by 2 to get from one number to the next. Since the first number is 2 the formula, there isn t any algebra needed to find a number to put in front of the 2. Answer: a n = 2 n

4 Section 9.1: Sequences 37) 6,12,24,48 Each number is a multiple of 2 apart. I need a 2 n in the formula. I will need a number in front of the 2 n to get the correct answer. I find the number by solving x(2 1 ) = 6 x = 3 Answer: a n = 3(2 n ) 39) 1,2,4,8,16 Each number is a multiple of 2 apart. I need a 2 n in the formula. I will need a number in front of the 2 n to get the correct answer. I find the number by solving x(2 1 ) = 1 x = 1/2 Answer: a n = (2 n ) it would also be correct to write 2 n-1 #41-48: Write the first 3 terms of the recursively defined sequence. 41) a 1 = 6, a k+1 = a k + 3 a 2 = a a 2 = a 2 = 9 a 3 = a a 3 = a 3 = 12 Answer: first 3 terms 6,9,12

5 Section 9.1: Sequences 43) a 1 = 6, a k+1 = 2(a k 1) a 2 = 2(a 1 1) a 2 = 2(6-1) a 2 = 10 a 3 = 2(a 2 1) a 3 = 2(10-1) a 3 = 18 Answer: first three terms are 6,10,18 45) a 1 = -3, a k+1 = 2a k + 3 a 2 = 2a a 2 = 2*(-3) + 3 a 2 = -3 a 3 = 2a a 3 = 2(-3) + 3 a 3 = -3 Answer: first three terms -3,-3,-3 47) a 1 = 6, a k+1 = a k + 4 a 2 = 7 Answer: First three terms 6,7,

6 Section 9.1 sequences #49-56: find the indicated sum 49) = 2*1 + 2*2 + 2*3 + 2*4 = Answer: 20 = (2*2-3) + (2*3 3) + (2*4 3) + (2*5 3) = Answer: 16 = = Answer: 31 = (-1) 4 + (-1) 5 + (-1) 6 + (-1) 7 +(-1) 8 = 1 + (-1) (-1) + 1 Answer: 1 = Answer:

7 Section 9.1 sequences = Answer: 30

8 Section 9.2: Arithmetic sequences: #1-8: show that each sequence is arithmetic. Find the common difference and write out the first 4 terms. 1) a n = 3n + 5 a 1 = 3(1) + 5 = 8 a 2 = 3(2) + 5 = 11 a 3 = 3(3) + 5 = 14 a 4 = 3(4) + 5 = 17 Answer: common difference = 3 first 4 terms; 8,11,14,17 3) a n = 5n - 2 a 1 = 5(1) 2 = 3 a 2 = 5(2) 2 = 8 a 3 = 5(3) 2 = 13 a 4 = 5(4) 2 = 18 Answer: common difference = 5 first 4 terms; 3,8,13,18 5) a n = 3 4n a 1 = 3-4(1) = -1 a 2 = 3 4(2) = -5 a 3 = 3 4(3) = -9 a 4 = 3-4(4) = -16 Answer: common difference = -4 first 4 terms; -1,-5,-9,-16

9 Section 9.2: Arithmetic sequences: 7) a n = 4 6n a 1 = 4 6(1) = -2 a 2 = 4 6(2)= -8 a 3 = 4 6(3) = -14 a 4 = 4 6(4) = -20 Answer: common difference = -6 first 4 terms; -2,-8,-14,-20 #9-18: Find a formula for the n th term of the arithmetic sequence with initial term a 1 and common difference d. 9) a 1 = 4; d = 3 I know I need a 3n for the common difference to be 3. I just need to solve for the number after the 3n. a n = 3n+x a 1 = 3(1) +x 4 = 3 + x 1 = x Answer: a n = 3n+1 11) a 1 = -2; d = 1 I need a 1n, for the common difference of 1. I need to solve for the number after the 1n. a n = 1n+x a 1 = 1(1) + x -2 = 1 + x -3 = x Answer: a n = 1n 3 or just n - 3

10 Section 9.2: Arithmetic sequences: 13) a 1 = 4; d = -3 I need a -3n, for the common difference of -3. I need to solve for the number after the -3n. a n = -3n+x a 1 = -3(1) + x 4 = -3 + x 7 = x Answer: a n = -3n ) a 1 = -2; d = -1 I need a -1n, for the common difference of -1. I need to solve for the number after the -1n. a n = -1n+x a 1 = -1(1) + x -2 = -1 + x -1 = x Answer: a n = -1n 1 or just -n - 1 #17-24: Find the indicated term in each arithmetic sequence. 17) 50 th term of, 3,6,9,12 First I need a formula for a n ; a n = 3n + x a 1 = 3(1) +x 3 = 3 + x 0 = x Formula for sequence: a n = 3n computation of 50 th term: a 50 = 3(50) = 150 Answer: 150

11 Section 9.2: Arithmetic sequences: 19) 100 th term of 5, 8, 11, 14 First I need a formula for a n : a n = 3n + x a 1 = 3(1) +x 5 = 3 + x 2 = x Formula for sequence: a n = 3n +2 computation of 100 th term: a 100 = 3(100)+2 = 302 Answer: ) 80 th term of 5,2,-1,-4 First I need a formula for a n : a n = -3n + x a 1 = -3(1) +x 5 = -3 + x 8 = x Formula for sequence: a n = -3n + 8 computation of 80 th term: a 80 = -3(80)+8 = -232 Answer: ) 25 th term of 20, 18,16,14 First I need a formula for a n : a n = -2n + x a 1 = -2(1) +x 20 = -2 + x 22 = x Formula for sequence: a n = -2n +22 computation of 25 th term: a 25 = -2(25)+22 Answer: -28

12 Section 9.2: Arithmetic sequences: 25) A local theater has 30 seats in the first row and 50 rows in all. Each successive row contains two additional seats. How many seats are in the 50 th row of the theater? This is just a arithmetic sequence problem with a 1 = 30, and d = 2. I need to find a 50. First I will get a formula for a n a n = 2n+x a 1 = 2(1) + x 30 = 2 + x 28 = x Formula for sequence: a n = 2n+28 now to find the number of seats in the 50 th row: a 50 = 2(50) + 28 = 128 Answer: There are 128 seats in the 50 th row. 27) An outdoor amphitheater has 40 seats in the first row, 41 seats in the second row, 42 seats in the third row. The pattern continues. The amphitheater has 40 rows. How many seats are in the 30 th row of the theater? This is just a arithmetic sequence problem with a 1 = 40, and d = 1. I need to find a 40. First I will get a formula for a n a n = 1n+x a 1 = 1(1) + x 40 = 1 + x 39 = x Formula for sequence: a n = n+39 now to find the number of seats in the 40 th row: a 40 = = 79 Answer: There are 79 seats in the 40 th row.

13 Section 9.2: Arithmetic sequences: #29 40 Find the indicated sum I can write out the problem without summation notation as follows; 2(1) + 2(2) + 2(3) +.+ 2(40) This is asking me to add the first 40 terms of an arithmetic sequence. I will use the formula S n = (a 1 + a n ) In this case: S 40 = =20(82)=1640 Answer: 1640 I can write out the problem without summation notation as follows; (2*1-3) + (2*2-3) + (2*3 3) +.+ (2*60-3) This is asking me to add the first 60 terms of an arithmetic sequence. I will use the formula S n = (a 1 + a n ) In this case: S 60 = =30(116)=3480 Answer: 3480

14 Section 9.2: Arithmetic sequences: I can write out the problem without summation notation as follows; (5-2*1) + (5-2*2) + (5-2*3) +.+ (5-2*40) (-1)+. + (-75) This is asking me to add the first 40 terms of an arithmetic sequence. I will use the formula S n = (a 1 + a n ) In this case: S 40 = =20(-72)= Answer: I can write out the problem without summation notation as follows; (60-5*1) + (60-5*2) + (60-5*3) +.+ (60-5*20) (-40) This is asking me to add the first 20 terms of an arithmetic sequence. I will use the formula S n = (a 1 + a n ) In this case: S 20 = =10(15)=150 Answer: 150

15 Section 9.2: Arithmetic sequences: 37) A local theater has 30 seats in the first row and 50 rows in all. Each successive row contains two additional seats. How many seats are in the theater? I need to know how many seats are in the 50 th row. I will find a formula for a n to get started. a n = 2n + x a 1 = 2(1) + x 30 = 2 + x 28 = x formula for sequence: a n = 2n + 28 computation of number of seats in 50 th row: a 50 = 2(50) + 28 = 128 I need to add these numbers to get my answer: I will use the formula S n = (a 1 + a n ) S 50 = Answer: There are 3950 seats. 39) An outdoor amphitheater has 40 seats in the first row, 41 seats in the second row, 42 seats in the third row. The pattern continues. The amphitheater has 70 rows. How many seats does the amphitheater contain? I need to know how many seats are in the 70 th row. I will find a formula for a n to get started. a n = 1n + x a 1 = 1(1) + x 40 = 1 + x 39 = x formula for sequence: a n = n + 39 computation of number of seats in 70 th row: a 70 = = 109 I need to add these numbers to get my answer: I will use the formula S n = (a 1 + a n ) S 50 = Answer: There are 5215 seats

16 Section 9.3: Geometric Sequences and Series: #1-10: List the first 4 terms of the geometric series and find the common ratio (r). 1) a n = 2 n a 1 = 2 1 = 2 a 2 = 2 2 = 4 a 3 = 2 3 = 8 a 4 = 2 4 = 16 Each term is a multiple of 2 apart so the common ratio is 2. Answer: First 4 terms 2,4,8,16 common ratio 2 3) a n = 3(2 n ) a 1 = 3(2 1 ) = 6 a 2 = 3(2 2 ) = 12 a 3 =3( 2 3 ) = 24 a 4 = 3(2 4 ) = 48 Each term is a multiple of 2 apart so the common ratio is 2. Answer: First 4 terms 6,12,24,48 common ratio 2 5) a n = ( ) ( ) ( ) ( ) ( ) Each term is a multiple of ½ apart so the common ratio is Answer: first 4 terms common ratio ½

17 Section 9.3: Geometric Sequences and Series: 7) a n =2 3n a 1 = 2 3*1 = 8 a 2 = 2 3*2 = 64 a 3 = 2 3*3 = 512 a 4 = 2 3*4 = 4096 Each term is a multiple of 8 apart so the common ratio is 8. Answer: first 4 terms 8,64,512,4096 common ratio 8 9) a n = 2(3 2n ) a 1 = 2(3 2*1 ) = 18 a 2 = 2(3 2*2 ) = 162 a 3 = 2(3 2*3 ) = 1458 a 4 = 2(3 2*4 ) = Each term is a multiple of 9 apart so the common ratio is 9. Answer: first 4 terms 18,162,1458,13122 common ratio 9 #11-18: Find a formula for the n th term of the geometric sequence with first term a 1 and common ratio (r). 11) a 1 = 2; r = 3 a n = x(3 n ) a 1 = x(3 1 ) 2 = 3x Answer: a n = 13) a 1 = 6; r = 2 a n = x(2 n ) a 1 = x(2 1 ) 6 = 2x 3 = x Answer: a n = 3(2 n )

18 Section 9.3: Geometric Sequences and Series: 15) a 1 = 1; r = a n = ( ) 1 = ( ) 1 = 2*1 = 2* 2 = x Answer: a n = ( ) or 2 1-n 17) a 1 = 4; r = a n = x( ) a 1 = x( ) 4 = 2*4 = 2* 8 = x Answer: a n = 8( ) #19-26: Find the indicated term of the geometric sequence 19) The 10 th term; 6,18,54,162, First develop a formula for a n ; first term is 6 and common ratio is 3 a n = x(3 n ) a 1 = x(3 1 ) 6 = x(3 1 ) 2 = x formula for a n = 2(3 n ) Now find the 10 th term: a 10 = 2(3 10 ) = Answer:

19 Section 9.3: Geometric Sequences and Series: 21) The 9 th term; 5,20,80,160 First develop a formula for a n ; first term is 5 and common ratio is 4 a n = x(4 n ) a 1 = x(4 1 ) 5 = 4x = x formula for a n = Now find the 9 th term: a 9 = = Answer: ) The 14 th term; 200,100,50,25 First develop a formula for a n ; first term 200, common ratio ½ a n = x( ) 200 = x( ) 200 = 2*200 = 2* 400 = x formula for a n = 400( ) now find the 14 th term: a 14 = 400( ) Answer:

20 Section 9.3: Geometric Sequences and Series: 25) The 20 th term; 39366, 13122, 4374, First develop a formula for a n ; first term common ratio a n = x( ) = x( ) = 3*39366 = 3* = x a n = ( ) now I can find a 20 = ( ) = Answer: 27) Jess currently has a salary of $40,000 per year. She expects to get an annual raise of 2%. What will her salary be when she begins her 10 th year? First term of sequence 40,000 common ratio 1.02 formula for a n = 40000(1+02) n a 10 =40000(1.02) 10 = Answer: $48, ) The price of the average Phoenix home is $135,000. The price is expected to increase by 2.5% per year. What will the average price of a Phoenix home be in 8 years? a n = (1+025) n a 8 = (1.025) 8 = Answer: $