Sequences and Series Concepts and Problems, Summaries of Mathematics

Download Sequences and Series Concepts and Problems and more Summaries Mathematics in PDF only on Docsity! IB Standard Level Analysis & Approaches Analysis and Approaches Sequences & Series (C) ¨ Arithmetic ¨ Geometric ¨ Sum to Infinity ¨ Sigma Notation ¨ Compound Interest Arithmetic seavence t or V6 3 5 7 9 11 13 E d uz 01 5 3 2 T 5 nth term formula Un u ten e d ai so Icu toniii V10 37 10 112 56 3 9 X2 3 3 13 3 16 48 Iii Sns I quit in Dd Size Size 2133 11 2 6 6 22 6 28 16811 Geometric sequence x or 4 8 16 32 64 u uz r 4 2 n un v x r n sn u lI V5 4 125 or 4 2 s v.li v21 4 16 41,311 124 IB Standard Level Analysis & Approaches 3 Geometric Sequence 13. For each sequence below, use the general formula to find the term indicated. a) 5, 10 20, … (* b) 2, #$ , # #$ , … ($ 14. Find the sum of the following geometric sequence. a) −2 + 8 − 32 + …+ (" b) & !' + ! $ + # !$ + …+ (" 15. Find the number of terms in each of these geometric sequences. a) 2, 10, 50, … 1250 b) 4,−12, 36, …− 972 16. Find the sum of each of these geometric sequences. a) 3 + 6 + 12 + …+ 384 b) 1 + ! # + ! ( + …+ ! #! 17. Find 4 given that the following are consecutive terms of a geometric sequence. a) 7, 4, 28 b) 4, 34, 20 − 4 18. A geometric sequence has 3rd term 75 and 4th term 375. Find the common ratio and the first term. 19. Find the sum of the first ten terms of a geometric sequence, whose 3rd term is 20 and 8th term is 640. Sn i x III 2 i g I v3 u x o 20 Us v xv7 Gyo 85 32 8 2 IB Standard Level Analysis & Approaches 4 20. In a geometric sequence the sum of the 2nd and 3rd term is 12, and the sum of the 3rd and 4th term is 60. Find the common ratio. 21. Alexa wins a prize in a lottery. She receives CAD 8000 the first month, CAD 6000 the second month, CAD 4500 the third month and so on for a total of six months. a) Calculate how much she receives in month six. b) Calculate the total prize money for the six months. 22. Beau spends $15000 buying computer materials for his office. Each year the material depreciates by 12%. a) Find the value of the material after three years. b) Find how many years it takes for the materials to be worth $5000. 23. A nest of ants initially contains 500 individuals. The population is increasing by 12% each week. a) How many ants will there be after: i) 10 weeks ii) 20 weeks? Sum to Infinity 24. Find the sum of each of the following infinite geometric series: a) 100 + 10 + 1 + … b) 20 + 10 + 5 + … 25. Write the following as a rational number: a) 0. 4; b) 0. 16;;;; So I I L t a 1 r t so Ea IB Standard Level Analysis & Approaches 6 Answers 1. a) –25 b) $ # 2. a) 725 b) –2775 3. a) 21 b) 44 4. a) 35.4 b) 120 5. a) 17.5 b) 4 c) 3, –2 6. a) 1.5 b) 11 7. a) (! + 27 = 12, (! + 97 = 40 b) (! = 4, 7 = 4 c) 400 8. 664 9. (! = −11, 7 = 3 10. a) 8) = ) # ?8 − 3(9 − 1)@ b) –95 c) 15 11. a) 7.5 b) 97.5 12. 4752 13. a) 640 b) # "#$ 14. a) 1638 b) 0.821 15. a) 5 b) 6 16. a) 765 b) 2 − =!#> ) 17. a) ±14 b) 2 18. B = 5, (! = 3 19. U1=5, S10=5115 IB Standard Level Analysis & Approaches 7 0. 5 1. a) CAD 1898.44 b) CAD 26 304.69 22. a) $10 222.08 b) 8.59 years 23. a) i) 1550 ii) 4820 b) 13.2 24. a) !''' / b) 40 25. a) ( / b) !" // 26. 12 27. # & 28. 48 ft 29. a) 50 b) 1364 c) ! ( 30. $51 249.06 31. $11 477.02 32. Ryan, $16.31 more 1 IB Standard Level Analysis and Approaches Sequences & Series(H)  Arithmetic  Geometric  Sum to Infinity  Sigma Notation  Compound Interest 4 IB Standard Level 14. Pedro is bored and is making patterns with sticks. Pattern 1 was made with four sticks, pattern 2 with seven sticks and so on. a) Write down the number of sticks needed to make pattern 5 and pattern 6. b) Find the number of sticks needed to make pattern 20. c) Find the pattern number for the pattern made with 127 sticks. 15. For each sequence below, use the general formula to find the term indicated. a) 1, −3, 9, … 𝑢7 b) 2, 3, 4 1 2 , … 𝑢9 c) 1, − 1 3 , 1 9 , … 𝑢6 16. Find the sum of the following geometric sequence. a) 5 + 10 + 20 + … + 𝑢8 b) 1 + 1 3 + 1 9 + … + 𝑢7 c) 1 2 + 1 4 + 1 8 + … + 𝑢𝑛 17. Find the number of terms in each of these geometric sequences. a) 3, 6, 12, … 768 b) 1, −2, 4, … 1024 e metric Sequence 5 IB Standard Level c) 54, 18, 6, … 2 27 18. Find the sum of each of these geometric sequences. a) 2 + 6 + 18 + … + 1458 b) 7 − 14 + 28 − … + 448 c) 1 3 − 1 9 + 1 27 − … − 1 729 19. Find 𝑘 given that the following are consecutive terms of a geometric sequence. a) 18, 𝑘, 40.5 b) 𝑘, 𝑘 + 8, 9𝑘 c) 7𝑘 − 2, 4𝑘 + 4, 3𝑘 20. In a geometric sequence and 2nd term is -12 and the 5th term is 768. Find the common ratio and the corresponding value of the 1st term. 21. In a geometric sequence, the 2nd term is 15 and the 5th term is -405. Find the sum of the first eight terms. 22. A geometric sequence is such that the sum of the 4th and 5th term is -108, and the sum of the 5th and the 6th terms is 324. Calculate the common ratio and the value of the 1st term. 23. Glovanna works on a farm and feeds the chickens each week. At the beginning the farm has 500 chickens. The number of chickens increases each week by 1%. a) Find the number of chickens after 15 weeks. b) Find the total number of feeds in 15 weeks. 24. Petra buys a camper van for $45000. Each year the camper van decrease in value by 5%. Find the value of the camper van at the end of six years. 6 IB Standard Level 25. A herd of 32 deer is to be left unchecked on a large island off the coast of Alaska. It is estimated that the size of the herd will increase each year by 18%. a) Estimate the size of the herd after: i) 5 years ii) 10 years b) How long will it take for herd size to reach 5000? 26. Two species of spiders inhabit in a remote island. The population of species A is 12000 and is increasing at a rate of 1.25% per month. The population of species B is 50000 and is decreasing at a rate of 175 spiders each month. When will the population of species A be greater than the population of species B? 27. Find the sum of each of the following infinite geometric series: a) 64 + 16 + 4 + … b) √3 + √3 4 + √3 16 + … 28. Write the following as a rational number: a) 0.3̅̅ ̅̅ b) 0. 27̅̅̅̅ 29. The sum of an infinite geometric series is 20, and the common ratio is 0.2. Find the first term of this series. 30. The sum of the first 3 terms of a convergent infinite geometric series is 19. The sum of the series is 27. Find the first term and the common ratio. 31. The first 3 terms of an infinite geometric sequence are 𝑚 − 1, 6, 𝑚 + 8. a) Write down two expressions for 𝑟. Sum t n init