8.1-8.2 In Class Review Solution Key

HPC 8.1-8.2 In Class Review Sequences & Series [May 2016] Name: What patterns should you look for when presented with a sequence? #adjust any of these by alternating 1. arithmetic 2. geometric 3. squares 4. cubes 5. powers 6. factorial 7. How do you know a sequence is arithmetic? common difference (constant difference) between terms, graph (and equation) is linear 8. What is the rule for an arithmetic sequence? An = a1 + d(n-1) 9. What is the rule for the sum of an arithmetic sequence? Sn = n/2(a1 + an) or Sn = n/2[2a1 + d(n-1)] 10. If you're missing any terms to find the sum of an arithmetic sequence, where should you go to find what's missing? an = a1 + d(n-1) Find an expression for the apparent nth term of the sequence. 11. 17, 13, 9, 5, ... an = -4n + 21 an = 17 + (-4)(n-1) an = 17 - 4n + 4 an = -4n + 21 12. 1/3, 4/5, 9/7, 16/9, ... an = n² / (2n+1) numerator: perfect squares denominator: arithmetic 2n+1 13. 2, -9, 28, 65, 126, ... an = (-1)^(n+1)(n³+1) Alternating... one more than perfect cubes 14. 1/7, 1/12, 1/19, 1/28, ... an = 1/(n²+2) numerator: constant denominator: two more than perfect squares 15. 0, 1/2, 2/6, 3/24, 4/120, ... an = (n-1)/n! numerator: arithmetic 7-1 denominator: factorial n! Simplify the following (by hand). 16. 10!/7! = 120 17. 2n!/(n-1)! = 2n 18. In an arithmetic sequence, a11 = 11 and a14 = 39. Find a67. (39 - 11)/(14 - 4) = 7 a67 = a1 + 7(67 - 1) a67 = -10 + 7(66) a67 = 452 Give a recursive definition for the following sequences. Don't forget to state the first term! The formula means NOTHING without 1   1. it! 19. 13, 9, 5, 1, -3, ... a1 = 13 an = an-1 - 4 20. 14, -28, 56, -112, 224, ... a1 = 14 an = -2an-1 Find the following sums. 21. 2 + 6 + 10 + 14 + ... + 398 + 402 402 = 2 + 4(n - 1) 101 = n (101/2)(2 + 402) = 20,402 22. Sum of the first 100 positive odd integers an = 1 + 2(n - 1) an = 2n - 1 a1 = 1 a100 = 199 (100/2)(1 + 199) = 10,000 23. Σ(n³ - 1) from n = 1 to 17 * Not arithmetic, no application of "rule" * Calculator: 23,392 24. Sum of the first 100 positive multiples of 5 an = 5 + 5(n - 1) an = 5n n = 100, a1 = 5, a100 = 500 (100/2)(5 + 500) = 25,250 25. Σ(8n - 5) from n = 1 to 8 8(1) - 5 + 8(2) - 5 + 8(3) - 5 + 8(4) - 5 + 8(5) - 5 + 8(6) - 5 + 8(7) - 5 = 189 26. Σ(81 - 5) from n = 1 to 2 "-5" only happens at the end 81(2) - 5 = 157