University:
Winston-Salem Forsyth County Schools
Course:
AP Calculus AB
Academic year:

2014
Views:

498

Pages:

3
Author:

Meadow Benjamin

Key Arithmetic: (u_{n} = u_{1} + d(n-1)) (S_{n} = \frac{n(u_{1} + u_{n})}{2}) Geometric: (u_{n} = u_{1}(r)^{n-1}) (S_{n} = \frac{u_{1}(1-r^n)}{1-r}) 1. Write a recursive formula for each sequence. a. 2, 5.6, 13.2, 20.8, ... (u_{n} = u_{n-1} + 7.6) b. 5, 8.5, 14.45, -24.565, ... (u_{n} = 1.5u_{n-1} - 5) c. 2.5, -1, 25, -6.875, ... (u_{n} = -2.5u_{n-1} - 5) d. -7.3, 15.89, 32.123, 43.4861, ... (u_{n} = 0.7u_{n-1} + 21) 2. 163, 122.25, 91.6875, 68.765625, ... a. Find the common ratio. 0.75 b. Write a recursive formula for the sequence. (u_{n} = 0.75u_{n-1}) c. Identify the sequence as growth or decay. Decay d. Give the percent increase or decrease. 25% decrease e. Find U₃₅ (Round to the nearest hundredth). U₃₅ ≈ 0.0092 3. Find the long-run value (limit). a. uₙ = 7.5 * 0.5ⁿ 0 b. uₙ = 0.2x + 40 50 5. -1.2 + 2.6 + 6.4 + 10.2 + 14 + ... a. Write a recursive formula to model the sequence. (u_{n} = u_{n-1} + 3.8) b. Write an explicit formula for the sequence. (u_{n} = -1.2 + 3.8(n-1)) c. Find U₁₂. 116.6 d. Find S₃₈. 38(-1.2 + 139.4) / 2 = 2625.8 e. Find Σ₂⁵ (25 - 2(1.0 + 11.2)) 25(-8.2 + 101) / 2 = 1160 6. Find the indicated value. a. U₃ if uₙ = -2 + 5n 13 b. U₄₅ if uₙ = -2(1.3)ⁿ -385.07 c. S₂₅ if uₙ = -1 + 1.5n 25(-0.5 + 36.5) / 2 = 450 d. S₂₃ if uₙ = 1.4(1.2)ⁿ⁻¹ 456.73 e. Σₙ=₆³⁰ (5 - 2.2n) 25(-8.2 + 101) / -0.5 = -8805 f. Σₙ=₁∞ 1.8(0.6)ⁿ⁻¹ 1.8 / (1 - 0.6) = 4.5 g. Σₙ=₁∞ (-2)(1.5)ⁿ⁻¹ -2 / (1 - 1.5) = -4 h. Σₙ=₉¹⁰⁰ (9n + 10) 94446 i. Σₙ=₁₄ (12.4 + 3.8n) 1666.6 j. Σₙ=₁⁴ (1.4 + 2.8 - 5.6 + ... + 11744051.2) 7829367 Image Analysis Overall Description: The image is a handwritten worksheet containing problems related to sequences and series, specifically arithmetic and geometric sequences. It includes exercises on writing recursive formulas, finding specific terms, calculating sums, and determining convergence. Textual Content: 7. The first term of a geometric sequence is 4 and the sum of the series is 12. Find the common ratio and list the first five terms of the sequence. Solution: (S = 4 + 4r + 4r² + ... = 12) (S = \frac{4(1 - r^n)}{1 - r} = 12) (12 - 12r = 4) (r = \frac{2}{3}) First five terms: 4, 8/3, 16/9, 32/27, 64/81 8. Marcy drops a ball from a height of 15 feet. On each bounce, the ball returns to 70% of its previous height. What is the total distance the ball travels before coming to rest? Solution: (S = 15 + 15(0.7) + 15(0.7)² + ...) (S = \frac{15}{1 - 0.7} = 50) feet 9. Samantha was hired as a first-year teacher at an annual salary of $32,000. She will receive an annual salary increase of 4% each year. a. Write an explicit formula to model the situation. (U_{n} = 32,000(1.04)^{n-1}) b. How much will she be making in her 5th year? $37,435.47 c. How much total will she make during her first 10 years of teaching? (S = \frac{32,000(1 - 1.04^{10})}{1 - 1.04} = $384,195.43) 10. Peter collects rocks. The first day he collected 8 rocks and each day after he collected 4 more than he did the day before. a. Write an explicit formula to model the situation. (U_{n} = 8 + 4(n-1)) b. How many rocks will he be collecting on the 10th day? 44 c. How many total rocks will Peter have after 20 days? (S = \frac{20(8 + 44)}{2} = 920)