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Sequences & Series

Explore arithmetic, geometric, and harmonic sequences and series, and learn how to find terms, sums, and apply sensitivity analysis. Also, discover the Fibonacci sequence, intelligence testing, numerology, and differential equations.

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Sequences & Series

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  1. Sequences & Series • A sequence is a progression of numbers: 3, 10, 19, 37, …In an arithmetic sequence the terms have a common difference: 1, 4, 7, 11, …. • In an harmonic sequence the terms are reciprocals of the terms in an arithmetic sequence: 1, 1/4, 1/7, 1/11, …. • In a geometric sequence the terms have a common ratio: 1, 3, 9, 27, …. • A series is the sum of the terms in a sequence 1 + 3 + 9 + 27 rd

  2. Arithmetic Sequences • Complete the sequences at the *a) 2 7 12 17 * * b) 5 13 21 * * c) 11 15 * 23 * • * * 20 29 38 e) 4 * 18 * 32 f) * 33 * 65 * • 10 * 70 h) 10 * * 70 i) 10 * * * * 70 • h) If each term of a sequence is multiplied by a constant, is the resulting sequence arithmetic? • a, a + d, a + 2d, a + 3d versus ka, k(a + d) k(a + 2d) k(a + 3d) rd

  3. Arithmetic Sequences • The 100th term of 2 5 8 11 14 * * * is ____. ans. 299 • b) The 20th term of 11 15 19 23 * * * is ____. ans. 87 • Find the sum of the sequence: 3 7 11 15 19 23 27 • Add 3 7 11 15 19 23 27 + 27 23 19 15 11 7 3 30 30 30 30 30 30 30 • => sum = 7(3 + 27)/2 = 7(30)/2 = 105 • d) Find the sum of the first 100 integers. rd

  4. Arithmetic Series Sum Sn of a finite arithmetic series is given by Sn = n(a1 + an)/2 Example: 2 + 4 + 6 + 8 + . . . + 100 = 50(2 + 100)/2 = 2550; where n = 100/2 = 50; a1 = 2; an = 100 1 + 5/3 + 7/3 + . . . + 201 = . 1 + (n – 1)(2/3) = 201 => n = 301 => Sn = 301(1 + 201)/2 = 30,401 rd

  5. Harmonic Series Arithmetic sequence 1 4 7 10 13 16 19 22 Reciprocals: 1 1/4 1/7 1/10 1/13 1/16 1/19 1/22is an harmonic sequence (+ 1 1/2 1/4 1/8 1/16 1/32 1/64 1/128 1/256 1/512 1/1024 1/2048 1/4096 1/8192 1/16384 1/32768 1/65536 1/131072 1/262144 1/524288 1/1048576)  2097151 / 1048576 = 1.9999999999… (let ((x 0)) (dotimes (i 100 x) (incf x (recip (expt 2 i))))) 2 rd

  6. Harmonic Series • Find the 36th term of the series • 1 + 1/4 + 1/7 + 1/10 + 1/13 + …The arithmetic series is 1 4 7 10 13 … • and the 36th term is 1 – 35*3 = 106 => 1/106. rd

  7. Harmonic Series • A cyclist travels from A to B at 40 mph and returns at 60 mph. The average speed for the round trip is . • 48 b) 49 c) 50 d) 51 e) none of these • (1 / [(1/40 + 1/60)/ 2] = 48 • Apply sensitivity analysis to explain why. rd

  8. Geometric Series Sum Sn of a geometric series of n terms is given by Sn = Find the sum of the geometric series 1 4 16 64 256 1024. Sum = (1 – 4 * 1024)/(1 – 4) = 1365 Find the sum of the geometric series 3 18 108 . . . 839,808. (3 – 6 * 839,808)/(1 – 6) = 1,007,769 rd

  9. Geometric Series • Find the sum of the following geometric series: • (1 + i)0 + (1 + i)1 + (1 + i)2 + (1 + i) 3 • Sum = (1 + i)0 - (1 + i)((1 + i)3 1 - (1 + i) • = 1 - (1 + i)4 -i (F/A, 6%, 4) • = (1 + i)4 – 1 i = F/A = [(1 + i)n – 1]/i = 4.37462 at i = 6% A A A A rd

  10. US Currency Bills: 1 2 5 10 20 50 100 500 1,000 5,000 10,000 100,000 Find a geometric sequence of bills with common ratio 10. 1 100 10,000 100,000 rd

  11. Pyramid Scheme rd

  12. Fibonacci Sequence (1 1 2 358 13 21 34 55 89 144 233 377 610987 1597 2584 4181 6765 10946) Which terms are evenly divisible by 3, 5, 8, 13 and 55? rd

  13. Intelligence Testing John is twice as old as his sister Mary, who is now 5 years of age. How old will John be when Mary is 30 years of age? rd

  14. Numerology • How much wheat can be put on a chessboard with 1 grain on the first square, 2 on the next, 4 on the third etc.? • S = 1 + 2 + 4 + 8 + 16 + 32 + … + 263 • (1 – 264)/(1 – 2) = • 18,446,744,073,709,551,615 grains of wheat • Roughly a train reaching a thousand times around the Earth. rd

  15. Differential Equation for e y’ – y = 0 Assume y = ex = a0 + a1x + a2x2 + a3x3 + a4x4 + a5x5+ . . .Then y’ = a1 + 2 a2 x + 3 a3 x2 + 4 a4 x3 + 5 a5 x4 + . . .and y(0) = 1 => a0 = 1 y’(0) = 1 => a1 = 1 y’’(0) = 1 => 2a2 = 1 => a2 ½ y’’’(0) = 1 => 6a3 = 1 => a3 = 1/6 ex = 1 + x + x2/2! + x3/3! + … rd

  16. Infinite Series Two motorcyclists A and B, 100 miles apart, head for each other. A travels at 40 mph and B at 60 mph. A fly flies from i’s nose to A’s nose and back again and again at 70 mph. How far will the fly have flown when the two cyclists meet? Infinite series whose terms increase in magnitude have no attainable sum. If a sum exists, series is said to be convergent; if not, divergent. rd

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