Math 11 Review Solution Key

Pre-Calculus 11.1-11.3, 11.6 Review Name: Find the first five terms of the sequence. 1. aₙ = 3n(n + 5) a₁ = 3(1)(1 + 5) = 18 a₂ = 3(2)(2 + 5) = 42 a₃ = 3(3)(3 + 5) = 72 a₄ = 3(4)(4 + 5) = 108 a₅ = 3(5)(5 + 5) = 150 2. aₙ = -5aₙ₋₁, a₁ = 3 a₂ = -5(3) = -15 a₃ = -5(-15) = 75 a₄ = -5(75) = -375 a₅ = -5(-375) = 1875 3. aₙ = (3aₙ₋₁ + 6) / 3, a₁ = -8 a₂ = (3(-8) + 6) / 3 = -6 a₃ = (3(-6) + 6) / 3 = -4 a₄ = (3(-4) + 6) / 3 = -2 a₅ = (3(-2) + 6) / 3 = 0 Find the sum of the sequence. 4. Σ(2k²) from k = 2 to 7 2(2²) + 2(3²) + 2(4²) + 2(5²) + 2(6²) + 2(7²) 8 + 18 + 32 + 50 + 72 + 98 = 278 5. Σ(3k² - 6k) from k = 0 to 5 3(0)² - 6(0) + 3(1)² - 6(1) + 3(2)² - 6(2) + 3(3)² - 6(3) + 3(4)² - 6(4) + 3(5)² - 6(5) = 75 6. Σ(1/k) from k = 1 to 4 1/1 + 1/2 + 1/3 + 1/4 = 25/12 Write the sum using sigma notation. Hint: Are any of these arithmetic or geometric? 7. 5/2 + 5/3 + 5/4 + ... + 5/17 Σ(5/(1+k)) from k = 1 to 16 8. 6 + 13 + 20 + 27 + ... + 97 97 = 6 + (n - 1)(7) n = 14 Σ(7k - 1) from k = 1 to 14 9. 3 - 9 + 27 - 81 + 243 - 729 Σ(3^k * (-1)^(k+1)) from k = 1 to 6 Determine whether the sequence is arithmetic or geometric, then find the nth term of the sequence. 10. 10, 8, 6, 4, 2, ... Arithmetic, d = -2 aₙ = 10 + (n - 1)(-2) aₙ = -2n + 12 11. 5, 1, 0.2, 0.04, ... Geometric, r = 1/5 aₙ = 5(1/5)^(n-1) 12. 1/4, 1/2, 1, 2, 4, ... Geometric, r = 2 aₙ = (1/4)(2)^(n-1) 13. -1, 1/2, -1/4, 1/8, ... Geometric, r = -1/2 aₙ = 1(-1/2)^(n-1) 14. 3.7, 4.3, 4.9, 5.5, 6.1, ... Arithmetic, d = 0.6 aₙ = 3.7 + (n - 1)(0.6) aₙ = 0.6n + 3.1 15. -6, -2, 2, ... Arithmetic, d = 4 aₙ = -6 + (n - 1)(4) aₙ = 4n - 10 Find the indicated term of the arithmetic sequence. 16. Find a₂₇ if a₁₀ = 115 and a₁₅ = 190. d = (190 - 115) / (15 - 10) = 15 190 = a₁ + (5 - 1)(15) a₁ = 100 aₙ = 100 + (n - 1)(15) a₂₇ = 100 + (27 - 1)(15) = 440 17. Find a₄₃ if a₃ = 13 and a₁₀ = 41. d = (41 - 13) / (10 - 3) = 4 41 = a₁ + (10 - 1)(4) a₁ = 5 aₙ = 5 + (n - 1)(4) a₄₃ = 5 + (43 - 1)(4) = 173 Write a rule for the nth term of the arithmetic sequence. 18. 85, 81, 77, 73, 94... d = 81 - 85 = -4 94 = a₁ + (3 - 1)(-4) a₁ = 100 aₙ = 100 + (n - 1)(-4) aₙ = 104 - 4n 19. 112, 115, 118, 61... d = 115 - 112 = 3 61 = a₁ + (21 - 1)(3) a₁ = 121 aₙ = 121 + (n - 1)(3) aₙ = 3n + 124 20. Which term of the arithmetic sequence -1, 4, 9, ... is 374? 374 = -1 + (n - 1)(5) 375 = 5(n - 1) 75 = n - 1 n = 76 21. Which term of the arithmetic sequence -0.3, 1.8, 3.9, ... is 1186.2? 1186.2 = -0.3 + (n - 1)(2.1) 1186.5 = 2.1(n - 1) 565 = n - 1 n = 566 Find the indicated term of the geometric sequence. 22. Find a₁₂ if a₆ = 3 and r = 2 3 = a₁(2)^(6-1) a₁ = 3/32 a₁₂ = (3/32)(2)^(12-1) a₁₂ = 192 23. Find a₂ if a₄ = 500 and r = 5 500 = a₁(5)^(4-1) a₁ = 4 a₂ = 4(5)^(2-1) a₂ = 20 Write a rule for the nth term of the geometric sequence. 24. a₃ = 12, a₁ = 3 12 = 3(r)^(3-1) 4 = r² r = ±2 aₙ = 3(±2)^(n-1) 25. a₈ = -8748, a₁ = 4 -8748 = 4(r)^(8-1) -2187 = r⁷ r = -3 aₙ = 4(-3)^(n-1) Find the partial sum Sₙ of the arithmetic sequence with the given conditions. 26. a₁ = 4, d = 3, n = 15 S₁₅ = 15(4 + (15-1)3)/2 = 255 27. a₁ = 12, d = 1/3, n = 13 S₁₃ = 13(2(12) + (13-1)(1/3))/2 = -182 Find the sum of the arithmetic series. 28. -6 + -2 + 2 + ... + 222 222 = -6 + (n-1)(4) 228 = 4(n-1) 57 = n-1 n = 58 S = 58(-6 + 222)/2 = 6264 29. Σ(11 + 2k) from k = 1 to 150 S = 150(13 + 311)/2 = 24300 30. Σ(7k - 3) from k = 1 to 300 S = 300(4 + 2097)/2 = 315,150 31. 16 + 10 + 4 + ... + (-50) -50 = 16 + (n-1)(-6) -66 = -6(n-1) 11 = n - 1 n = 12 S = 12(16 + (-50))/2 = -204