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1.1 ARITHMETIC SEQUENCES

1.1 ARITHMETIC SEQUENCES. A ____________________ is an ordered list of numbers in which each item follows another according to a pattern, such as {5, 8, 11, 14, ...}. It is said to be ____________________ when the pattern goes up or down by a constant amount.

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1.1 ARITHMETIC SEQUENCES

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  1. 1.1 ARITHMETIC SEQUENCES A ____________________ is an ordered list of numbers in which each item follows another according to a pattern, such as {5, 8, 11, 14, ...}. It is said to be ____________________ when the pattern goes up or down by a constant amount. This amount is referred to as d, or the ______________________________. Each item or term is denoted using tn, where n indicates the position in the sequence (t5 is the 5th term).

  2. 1.1 ARITHMETIC SEQUENCES Find the general term, or tn, for the sequence {5, 8, 11, ...}. For any arithmetic sequence, tn=

  3. 1.1 ARITHMETIC SEQUENCES FUNCTION NOTATION SEQUENCE NOTATION denoted by tn n can only be natural discrete Ex tn = 2n + 1 • denoted by f(x) • x can be any number • continuous Ex f(x) = 2x + 1

  4. 1.1 ARITHMETIC SEQUENCES Find tnand t10 for each of the following sequences. When is tn = 27? • {35, 33, 31, 29, ...} • {7, 12, 17, 22, ...} (SUBMIT ELECTRONICALLY)

  5. 1.1 ARITHMETIC SEQUENCES Determine the missing terms for each of the following arithmetic sequences. State the general term, tn. {_____, _____, 5, 9, 13, _____} {15, _____, _____, 9, _____} 292 is the _____ term in the sequence {-5, -2, 1, ...}

  6. 1.1 ARITHMETIC SEQUENCES Find tn for each of the following arithmetic sequences. a) t3 = 7, d = -3 b) t5 = -1, d = 5 (SUBMIT ELECTRONICALLY)

  7. 1.1 ARITHMETIC SEQUENCES Find tn for an arithmetic sequence if a) t5 = 10 and t8 = 16 b) t3 = -2 and t9 = 28 (SUBMIT ELECTRONICALLY)

  8. 1.1 ARITHMETIC SEQUENCES Find tn for an arithmetic sequence if the first three terms are a) {2x, 3x – 7, 5x + 2, ...} b) {5x + 1, 2x – 3, 7x + 9, ...} (SUBMIT ELECTRONICALLY)

  9. 1.2 ARITHMETIC SERIES An arithmetic series is the sum of the terms that make an _______________________________ . It is denoted by Sn, where n is the number of terms (S7 would be the sum of the first 7 terms of the series). If tn = 2n – 3 , find S5.

  10. 1.2 ARITHMETIC SERIES Find the sum of all the numbers from 1 to 100. For any arithmetic series, Sn=

  11. 1.2 ARITHMETIC SERIES Use the formula to find S19 when tn= -3n + 1 Use the formula to find S27 when tn= 2n – 5 (SUBMIT)

  12. 1.2 ARITHMETIC SERIES Find the sum 2 + 5 + 8 + 11 + ... + 68 Find the sum 3 + 8 + 13 + 18 + ... + 88 (SUBMIT ELECTRONICALLY)

  13. 1.2 ARITHMETIC SERIES Find the sum of the multiples of 6 from 10 to 100. Find the sum of the multiples of 4 from 10 to 100. (SUBMIT)

  14. 1.2 ARITHMETIC SERIES Find the sum of the multiples of 10 from 1 to 100, excluding the multiples of 4. Find the sum of the multiples of 4 from 1 to 100, excluding the multiples of 5. (SUBMIT ELECTRONICALLY)

  15. 1.2 ARITHMETIC SERIES Find S20 for each of the following. a) S3 = 45 and S6 = 135 b) S3 = 15 and S7 = 63 (SUBMIT ELECTRONICALLY)

  16. 1.2 ARITHMETIC SERIES Find S20 for each of the following. a) t7 = 11 and S7 = 35 b) t5 = -11 and S5 = -25 (SUBMIT ELECTRONICALLY)

  17. 1.3 GEOMETRIC SEQUENCES A sequence is said to be geometric when the pattern between successive terms involves ____________________ by the same value, such as {5, 15, 45, ...} and {32, 16, 8, ...}. This value is referred to as r, or the ______________________________. Find t8 for both examples above.

  18. 1.3 GEOMETRIC SEQUENCES Find the general term, tn , for {5, 15, 45, ...} . For any geometric sequence, tn=

  19. 1.3 GEOMETRIC SEQUENCES Determine tnand t20 for the following geometric sequences. • {3, 6, 12, 24, ...} • {2, 6, 18, 54, ...} (SUBMIT ELECTRONICALLY)

  20. 1.3 GEOMETRIC SEQUENCES Determine the general term, tn , for the following. How much is each car worth after 20 years? • You purchase a car for $20 000 and its value is reduced by 5% each year. • You purchase a car for $10 000 and its value is reduced by 8% each year. (SUBMIT ELECTRONICALLY)

  21. 1.3 GEOMETRIC SEQUENCES Determine the general term, tn , for the following. How much is each car worth after 20 years? • Your antique car, presently worth $20 000, increases in value by 4% each year. • Your antique car, presently worth $25 000, increases in value by 5% each year. (SUBMIT ELECTRONICALLY)

  22. 1.3 GEOMETRIC SEQUENCES Find the first four terms in each geometric sequence if a) t1 = 10 and r = 1.1 b) t1 = -2 and r = 2 (SUBMIT ELECTRONICALLY)

  23. 1.3 GEOMETRIC SEQUENCES A ball is dropped from a height of 6m. After each bounce it rises to 60% of its previous height. a) What is the height reached after the first bounce? b) State the general term, tn. c) What is the height reached after the 10th bounce? d) After how many bounces will the ball reach a height of approximately 17cm?

  24. 1.3 GEOMETRIC SEQUENCES Find tn for a geometric sequence if a) t5 = 10 and t8 = 17.28 b) t3 = -2 and t6 = 54 (SUBMIT ELECTRONICALLY)

  25. 1.4 GEOMETRIC SERIES A geometric series is the sum of the terms that make a _______________________________ . A ____________________ is a geometric figure created by repeating a pattern over and over, with each repetition referred to as an _______________________. They are said to be _______________________ when they contain smaller copies of themselves buried within.

  26. 1.4 GEOMETRIC SERIES Begin with a square of side length 16cm. Divide the square into quarters creating four smaller squares. Shade in one of these squares. This is the first iteration for this fractal. Show 5 iterations, calculating the area of each shaded square. a) Does the area of each successive square create a geometric sequence? b) Determine the total area shaded after five iterations.

  27. 1.4 GEOMETRIC SERIES Derive Sn for any geometric series. For any geometric series, Sn =

  28. 1.4 GEOMETRIC SERIES Find S10 for each geometric series. a) 5 + 15 + 45 + ... b) t1 = 3, r = 2 (SUBMIT ELECTRONICALLY)

  29. 1.4 GEOMETRIC SERIES Derive an alternate formula for Snthat doesn’t use the value of n. For any geometric series, Sn =

  30. 1.4 GEOMETRIC SERIES Find the sum of each geometric series. a) 5 + 10 + 20 + ... + 640 b) -2 + 6 + -18 + ... + 4374 (SUBMIT ELECTRONICALLY)

  31. 1.4 GEOMETRIC SERIES Find the value of t1 of each geometric series. a) Sn = -129, tn = -192, r = -2 b) Sn = 3124, tn = 2500, r = 5 (SUBMIT ELECTRONICALLY)

  32. 1.4 GEOMETRIC SERIES Find the value of t1 of each geometric series. a) S5 = 682, r = 4 b) S4 = 15, r = -2 (SUBMIT ELECTRONICALLY)

  33. 1.4 GEOMETRIC SERIES Find the value of n of each geometric series. a) Sn = 2730, t1 = 2, r = 4 b) Sn = 364, t1 = 1, r = 3 (SUBMIT ELECTRONICALLY)

  34. 1.4 GEOMETRIC SERIES A singles ping pong tournament is being organized at school. There are 128 players and losers from each round are eliminated from the next round. What is the total number of games to be played for the entire tournament? What if there were 512 participants? (SUBMIT)

  35. 1.4 GEOMETRIC SERIES You drop a ball from a height of 6m. After each bounce, the ball reaches only 60% of its previous bounces height. What is the total distance that the ball has travelled as it hits the floor for the 5th time? Height: 10m, Ratio: 40%, Bounces: 7 (SUBMIT)

  36. 1.5 INFINITE GEOMETRIC SERIES Infinite geometric sequences and series goes on forever, indicated by ... at the end. {1, 0.5, 0.25, 0.125, ...} 1 + 2 + 4 + 8 + 16 + ... An infinite geometric series is said to be _________________ when the sequence of partial sums approaches a fixed value. 1 + 0.5 + 0.25 + 0.125 + ... It is said to be ___________________ when the sequence of partial sums does not approach a fixed value. 1 + 2 + 4 + 8 + 16 + ...

  37. 1.5 INFINITE GEOMETRIC SERIES What fixed value does the following geometric series approach? 1 + 0.5 + 0.25 + 0.125 + ... For any converging infinite geometric series, S∞ =

  38. 1.5 INFINITE GEOMETRIC SERIES If you continue with the fractal below, what is the total area that will be shaded? (side length of largest square is 16cm) Do the perimeters form an infinite geometric series? If so, find the sum? (SUBMIT ELECTRONICALLY)

  39. 1.5 INFINITE GEOMETRIC SERIES Write 0.53535353... as an infinite geometric series. Find the sum. Repeat with 0.616161... (SUBMIT ELECTRONICALLY)

  40. 1.5 INFINITE GEOMETRIC SERIES Write 0.5893434... as an infinite geometric series. Find the sum. Repeat with 0.67161616... (SUBMIT ELECTRONICALLY)

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