Precalculus by Richard Wright

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When you lie down, you will not be afraid; when you lie down, your sleep will be sweet. Proverbs‬ ‭3‬:‭24‬ ‭NIV‬‬‬

10-Review

Take this test as you would take a test in class. When you are finished, check your work against the answers. On this assignment round your answers to three decimal places unless otherwise directed.

  1. Write the first four terms of the sequence \(a_n = n!-n\).

    2. Write the explicit formula for the nth term.

  2. 256, 192, 144, 108, …
  3. 20, 14, 8, 2, …
  4. \(\frac{3}{1}, \frac{4}{4}, \frac{5}{9}, \frac{6}{16}\)
  5. Simplify \(\frac{3!n!}{4!(n-2)!}\)
  6. The 3rd term of an arithmetic sequence is –2 and the 10th term is –16. What is the 5th term?
  7. The 2nd term of a geometric sequence is 4374 and the 7th term is 576. What is the 5th term?

    8. Find the sum. Show your work.

  8. \(\displaystyle \sum_{k=1}^3 (k-1)^2\)
  9. \(\displaystyle \sum_{n=1}^{14} 2n^3\)
  10. \(\displaystyle \sum_{i=1}^{21} -3i + 1\)
  11. \(\displaystyle \sum_{n=1}^{15} 3(2)^{n-1}\)
  12. \(\displaystyle \sum_{n=1}^{∞} 5\left(\frac{3}{4}\right)^{n-1}\)
  13. Use mathematical induction to prove the sum formula \(-4 + -1 + 2 + 5 + \cdots + (3n - 7) = \frac{3}{2}n^2 - \frac{11}{2}n\)
  14. Use the binomial theorem to expand \(\left(3x + 2\right)^4\).
  15. Find the coefficient of the term \(x^3\) in \(\left(x + 4\right)^5\).
  16. Evaluate \(_8C_3\).
  17. Evaluate \(_8P_3\).
  18. How many different license plates can be be made if each is 2 letters followed by 2 digits?
  19. Six people are going to a concert and are sitting in the same row. Fred has a broken leg and has to sit on the aisle and one of his 2 sisters wants to sit next him. How many different sitting arrangements can there be?
  20. What is the probability that you will randomly guess the answers to 4 out of 4 T/F quiz questions?
  21. If two coins are flipped, what is the probability of getting 2 tails?
  22. What is the probability of drawing a face card or a diamond from a standard 52-card deck?

Answers

  1. 0, 0, 3, 20
  2. \(a_n = 256\left(\frac{3}{4}\right)^{n-1}\)
  3. \(a_n = -6n + 26\)
  4. \(a_n = \frac{n+2}{n^2}\)
  5. \(\frac{n(n-1)}{4}\)
  6. –6
  7. 1296
  8. 5
  9. 22,050
  10. –672
  11. 98,301
  12. 20
  13. Show work and final step is \(\frac{3k^2 - 5k - 8}{2}\)
  14. \(81x^4 + 216x^3 + 216x^2 + 96x + 16\)
  15. 160
  16. 56
  17. 336
  18. 67,600
  19. 48
  20. 0.0625
  21. 0.25
  22. ≈ 0.423