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Name:__________________________________________________ Date:_______________________ Period:______________________ UNIT 5 TEST REVIEW: SEQUENCES AND SERIES 1. Mrs. Inscoe has square placemats made up of wooden circles. She notices that the wooden circles are put in diagonal lines starting in a corner. The first diagonal “row” has 1 wooden circle. The second diagonal row has 2 wooden circles, the third row has 3, and so on. How many wooden circles does it take to make up half of her placemat if one half has 16 rows in it? 2. Large dogs have an average of 7 puppies in a litter. If each of those puppies has a litter of 7 puppies, and so on for each generation, how many puppies can trace their lineage back to the original dog after 5 generations? (Assume the first generation is the original litter of 7 puppies.) ANSWER: 136 wooden circles (use arithmetic series formula; n = 16, a1 = 1, d = 1) ANSWER: 19,607 puppies (use geometric series formula; n = 5, a1 = 7, r = 7) 3. Charlie collects baseball cards, and he gets new cards each month. He starts out with no cards, and in the first month he gets 5 cards. In the second month, he gets 10 cards; in the third month, he gets 15 cards, and so on. How many cards does he receive in the 24th month? 4. What is the difference in a25 and S25? ANSWER: 120 cards (use arithmetic sequence formula because we don’t want to know how many he gets total, just in the 24th month… n = 24, d = 5, a1 = 5) 5. How do you determine whether or not a given sequence is arithmetic, geometric, or neither? ANSWER: a25 is asking for the 25th term and wants us to use a SEQUENCE formula. S25 is asking for the SUM of the first 25 terms and wants us to use a SERIES formula. 6. Write an example of an arithmetic sequence, a geometric sequence, and a sequence that follows a pattern but is neither arithmetic nor geometric. ANSWER: Arithmetic—add or subtract the same number every time Geometric—multiply by the same number every time. ANSWER: many correct answers. follow the rules to the left. example: arithmetic: 3, 5, 7, 9 geometric: 2, 6, 18, 54… 7. Write an arithmetic sequence that has four means between 3 and 88. 8. Write a geometric sequence that has 3 means between 256 and 81. ANSWER: 3, 20, 37, 54, 71, 88 (use the arithmetic sequence explicit formula to solve for d, then use it to find the means) ANSWER: 256, 192, 144, 108, 81 (use the geometric sequence explicit formula to solve for r, then use it to find the means) 9. Do the following sequences converge or diverge? 10. Does the following sequence converge or diverge? How do you know? an = 3(0.9)n – 1 ANSWER: Converge (geometric sequence, r < 1) 11. Does the following sequence converge or diverge? an = (3n + 4)/n 12. Does the following sequence converge or diverge? an = 0.8an-1 + 5 ANSWER: converge (graph it in your calculator…gets close to horizontal) ANSWER: Converge (generate the first 8-10 terms to see they are getting closer to one number) 15 13. Find 2n 5 n 4 14. Find S8 of the sequence 2, 6, 18, 54, … ANSWER: 6560 (geometric series formula) ANSWER: 168 (use arithmetic series formula) 15. What is the 30th partial sum of the series 1 + 4 + 7 + … 6 16. Find n 3 7 n 2 ANSWER: 1335 (arithmetic series formula) ANSWER: 405 (plug in 2 – 6 and add them all together) 17. Find the sum of the infinite geometric series 4 + 7 + 18. Find 12.25 + …. 4(0.2) n 1 n 1 ANSWER: can’t do it because r > 1 19. Find the common difference of the arithmetic sequence where a1 = 6 and a31 = 276. ANSWER: d = 9 (use arithmetic sequence explicit formula) 21. Write the explicit and recursive formulas for the following sequence: 240, 60, 15, 3.75… ANSWER: 5 (infinite geometric series formula) 20. Write the explicit and recursive formula for the following sequence: 12, 25, 38, 51… ANSWER: explicit: an = 12 + (n – 1)13 recursive: a1 = 12, an = an-1 + 13 22. Write the first 4 terms of the sequence an = n2 + 6 ANSWER: 7, 10, 15, 22 (plug in 1 – 4 for n) ANSWER: explicit: an = 240(.25)n-1 recursive: a1 = 240, an = an-1(.25) 23. Write the first 3 terms of the following sequence: a1 = 5, an = an-1(-6) 24. Given the following recursive formula: a1 = 7, an = an-1 + 3 Write the explicit formula that defines the same sequence. ANSWER: 5, -30, 180 ANSWER: an = 7 + (n – 1)(3) 25. Given the following explicit formula: an = 3(½)n – 1 Write the recursive formula that defines the same sequence. 26. Find the sum of the first n terms of a geometric series with a1 = 5, an= 5120, and r = 2. ANSWER: 10235 (geometric series formula) ANSWER: a1 = 3, an = an-1(½) 27. What is the difference between a sequence and a series? What notation/words tell you that you are dealing with a series? ANSWER: sequence: list of numbers, series: sum of the numbers in a sequence. Sequence: a Series: S, sigma 29. What is the common ratio of the following sequence? 4, 10, 25, 62.5… ANSWER: r = 2.5 28. What is the common difference of the following sequence? 18.5, 15, 11.5, 8, 4.5… ANSWER: -3.5 = d 30. What is the 40th term of the sequence 100, 89, 78, 67,… ANSWER: -329 (use arithmetic sequence explicit formula)