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Exploring Arithmetic and Geometric Sequences: Concepts, Problems, and Solutions

Dive into the world of arithmetic and geometric sequences with various problems including sequence determination, recursive definitions, polynomial expansion, and probability calculations. This comprehensive guide covers essential mathematical concepts such as the summation of series, combinations, and permutations. Learn to apply sigma notation, understand the behavior of different sequences, and gain insights into population growth and probability in real-world contexts. Suitable for students and math enthusiasts looking to enhance their problem-solving skills.

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Exploring Arithmetic and Geometric Sequences: Concepts, Problems, and Solutions

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  1. Arithmetic Or Geometric? Expand This! Learn to Count What are the Odds? Pot Pourri 1pt 1 pt 1 pt 1pt 1 pt 2 pt 2 pt 2pt 2pt 2 pt 3 pt 3 pt 3 pt 3 pt 3 pt 4 pt 4 pt 4pt 4 pt 4pt 5pt 5 pt 5 pt 5 pt 5 pt

  2. Determine whether the sequence is Arithmetic or Geometric. List d or r.6, 4, 2, 0, -2, …

  3. Arithmeticd= -2

  4. Given an = -4n + 13, Find a17

  5. -55

  6. Write a recursive definition for the sequence defined by 2, 6, 18, …

  7. an= 2(3)n-1

  8. Use sigma notation to write the sum of the following: 1 + 1 + 1 + …. + 1 2(1) 2(2) 2(3) 2(10)Then find the sum

  9. Sum = 7381 5040

  10. If n6=3, n9=81, find r and n1

  11. r=3, n1=1 81

  12. Expand (x + y)4

  13. x4+4x3y+6x2y2+4xy3+y4

  14. Give the third term of the expansion of (x + 1)8

  15. 28x6

  16. What term contains a6b3 in the expansion of (2a + 3b)9

  17. 4th term

  18. Find the coefficient of the term that contains a6b3 in the expansion of (2a + 3b)9

  19. 145152

  20. Expand (x-7)7

  21. x7-49x6+1029x5 -12005x4 +84035x3 - 352947x2 +823543x -823543

  22. A lunch consists of 3 different kinds of salads, 3 different kinds of soup, and 5 different drinks. How many choices are there for lunch?

  23. 45

  24. From a group of six boys and five girls, two boys and three girls will be selected to attend a conference. In how many ways can the selection be made?

  25. 25

  26. Find the number of distinguishable permutation of letters MISSISSIPPI.

  27. 34650

  28. A bag contains 4 purple marbles and 3 white marbles. How many ways can you draw 2 marbles?

  29. 21

  30. Find the number of diagonals in an octagon.

  31. 20

  32. A card is drawn from a deck of cards. Find the probability that it is not a face card.

  33. 10/13

  34. Of 100 students, 32 are taking Calculus, 28 are taking French, and 12 are taking both. If a student is picked at random, what is the probability that the student is taking Calculus or French?

  35. .48

  36. If you draw one card at random from a deck of 15 cards numbered 1 through 15, what is the probability that the number you draw is divisible by 3 and odd?

  37. 1/5

  38. If P(A) = .18, what is P(A’)

  39. .82

  40. A bag contains 4 red marbles and 3 blue marbles. One marble is drawn at random and not replaced. Then a second marble is drawn. What is the probability that the first marble is blue and the second is red?

  41. 2/7

  42. Simplify the ratio:85!79!

  43. 3.14 x 1011

  44. Find the sum of the series:1 + 1 + 1 + …4 16 64

  45. 1/3

  46. Find the sum of the multiples of 7 from 7 to 175

  47. 2184

  48. Manchester has a population of 125,000 people now and the population is increasing 3% every year. What will be the population in 5 years?

  49. 140688

  50. Find an explicit definition for the nth term of the sequence 6, 11, 16, 21, …

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