Mathematical Sequences and Series Practice Questions

Chapter 3 Chapter 3- - Sequences and series Sequences and series

Arithmetic sequences (3.1) Arithmetic sequences (3.1)

The ?th term of an arithmetic sequence is given by ??= 9 2?. What is the common difference? 1 a) 2 b) 2 c) 9 d) 9

The ?th term of an arithmetic sequence is given by ??= 4 3?. The ?th term of the sequence is 56. Find the value of ?. 2 a) 21 b) 21 c) 20 d) 20

The first term of an arithmetic sequence is 6 and the common difference is 14. Find the 86th term of the sequence. 3 a) 1184 b) 1198 c) 1196 d) 1204

4 An arithmetic sequence has a 3rd term of 10 and a 5th term of 6. What is the ?th term? a) 4? + 14 b) 14 2? c) 4? d) 16 2?

5 An arithmetic sequence has a 6th term of 9 and a 12th term of 13. What is the ?th term? b) 2 a) 4? + 5 3? + 5 d) 2 3? +14 c) 4? + 13 3

Arithmetic series (3.2) Arithmetic series (3.2)

An arithmetic progression has ?3= 20 and ?5= 44. Find the sum of the first 10 terms. 1 a) 500 b) 1000 c) 400 d) 800

An arithmetic progression has ?3= 12 and ?5= 20. Find the sum of the first 20 terms. 2 a) 1680 b) 420 c) 840 d) 100

An arithmetic sequence has a 3rd term of 10 and a 5th term of 6. Which of the following would correctly work out the sum of the first 15 terms? 3 a) 15 b) 15(2 15 + 14 2 ) 2(2 14 + 14 2 ) d) 15 c) 15 2(2 15 + 16 4 ) 2(2 14 + 15 2 )

4 Find the sum of the first 8 terms in the following series: 1 + 6 + 11 + 16 + a) 41 b) 148 c) 68 d) 36

5 The ?th term of an arithmetic Sequence is given by ??= 3? + 7. The sum of the first ? terms is 2250. Find the value of ?. a) 18 b) 41.66 c) 36 d) 38

Geometric sequences (3.3) Geometric sequences (3.3)

The first term of a geometric sequence is 12 and the common ratio is 2.5. Find the 9th term of this sequence. 1 a) 18,311 b) 45,776 c) 7324 d) 1538

2 The first term of a geometric sequence is 8 and the 6th term is 60.75. Find the common ratio. a) 2.5 b) 1.5 c) 1.5 d) 2.5

3 Three consecutive terms of a geometric sequence are ? 1,12 and 10? + 8. Which of the following is a possible value of ?? a) 4 b) 3.8 c) 4.8 d) 3.8

Geometric series (3.4) Geometric series (3.4)

The first term of a geometric sequence is 6 and the common ratio is 3. Find the sum of the first 5 terms of the sequence. 1 a) 1452 b) 240 c) 726 d) 486

The first term of a geometric sequence is 32 and the common ratio is 0.5. Find the sum of the first 10 terms of the sequence. 2 a) 127.875 b) 0.0625 c) 63.875 d) 63.938

3 The sum of the first three terms of a geometric series is 3. If the first term is 4, find the value of ?. b) 1 a) 1 4 2 d) 1 c) 1 4 2

Sum to infinity (3.5) Sum to infinity (3.5)

1 Which of the conditions below needs to be true for a geometric series to converge? a) ? is an integer b) ? > 0 c) ? < 1 d) ? < 1

2 Which of the following would calculate the following sum: 12 + 4 +4 3+1 1 12+ 3+ a) 12 ? 1 1 3 b) 12 2 3 1 3 c) 12 d) 1 3 12

3 Which of the following would calculate the following sum: 2 + 1 +1 2+ 1 + 4 b) 1 a) 4 3 d) 4 c) 4 3

4 A geometric series is such that the first term is 27 and the fourth term is 8. Find the sum to infinity. a) 81 b) 150 d) c) 270

Sigma notation (3.6) Sigma notation (3.6)

1 Which of the following is the correct notation to sum the first 10 terms of the following sequence: 7, 10, 13, 16, ? b) ?=7 103? + 4 17? + 3 a) ?=1 c) ?=7 10? + 3 104? + 3 d) ?=1

2 Which of the following is the correct notation to sum the first 20 terms of the following sequence: 7, 13, 19, 25, ? b) ?=7 20? + 6 27? + 6 a) ?=1 c) ?=7 206? + 1 206? + 1 d) ?=1

3 Which of the following would correctly work out ?=2 152? + 3? a) 15 b) 15(2 5 + 14 2 ) 2(2 7 + 14 2 ) d) 14 c) 14 2(2 5 + 13 2 ) 2(2 7 + 13 2 )

4 Which of the following would ?? 2 3 correctly work out ?=1 a) b) 2 d) 3 c) 3 2

5 Which of the following would correctly work out ?? 1 2 ?=5 3 a) 1.5 b) 0.5 0.5 1.5 3 c) 0.5 d) 32 0.5 3 32

6 Which of the following would correctly work out ?=3 103 4?? a) 12(1 410) 1 4 b) 12(1 49) 1 4 d) 192(1 47) 1 4 c) 192(1 48) 1 4

Recurrence relations (3.7) Recurrence relations (3.7)

1 A sequence is defined by the recurrence relation ??+1= 3?? 7 and ?1= 1. What is the value of ?3? a) 11 b) 19 c) 4 d) 1

2 A sequence is defined by the recurrence relation ??+1= 3??+ 4 and ?1= 2. What is the value of ?3? a) 34 b) 21 c) 18 d) 13

3 A sequence is defined by the recurrence relation ??+1=1 and ?2= 15. What is the value of ?4? 3??+ 1 a) 21 b) 21 9 3 c) 3 d) 30

4 A sequence is defined by the recurrence relation ??+1= ???+ ? and ?1= 4. Express ?3 in terms of ? and ?. a) 4?2+ ?? + ? b) 4 + 2? c) 4?2+ ?2? d) 2? + ?

5 A fish farm starts with a stock of 6000 fish. Each Friday 28% of the fish are removed for sale and it is then restocked with 400 new fish. Let ?? represent the number of fish after restocking ? times. What is the recurrence relation that describes the situation after restocking? a) ??+1= 0.28??+ 400 and ?0= 6000 b) ??+1= 0.72??+ 400 and ?0= 6000 c) ??+1= 0.28(??+ 400) and ?0= 6000 d) ??+1= 0.72(??+ 400) and ?0= 6000

6 The population of rabbits in a breeding centre increases by 5% during each month. At the end of each month the breeder sells 30 rabbits. If ?? represents the rabbit population at the beginning of a month, find an expression for ??+1. a) ??+1= 1.5??+ 30 b) ??+1= 5?? 30 c) ??+1= 1.05?? 30 d) ??+1= 0.95?? 30

Modelling with series (3.8) Modelling with series (3.8)

1 Robert starts his new job on a salary of 15,000. He gets 1000 rise each year at the end of the year until he reaches the maximum salary of 25,000. Find his total earnings after 8 years. b) 22,000 a) 148,000 c) 296,000 d) 23,000

2 Jane has developed a savings plan that forms an arithmetic sequence. She will save 1 during the first week of the year, 4 in the second week, 7 in the third week, etc. Calculate the amount she will save in the final week of the year. a) 154 b) 153 d) 156 c) 157

3 A car depreciates in value by 15% a year. After 3 years it is worth 11,054.25. What was the car s initial price? b) 18,000 a) 15,300 c) 16,812 d) 14,619