Math Sequences and Series Test Review Key

Sequences and Series Algebra 2 Honors Test Review Name: Key Date: Decide whether the sequence is arithmetic, geometric, or neither. If it is arithmetic, state the value of d. If it is geometric, state the value of r. Lastly, find the next two terms of the sequence.   9, 16, 23, 30, ... Arithmetic, d = 7 37, 44 1, 2, 4, 7, ... Neither 1/2, 2/3, 3/4, ... Neither 500, 100, 20, 4, ... Geometric, r = 1/5 4/5, 4/25 Find the next two terms of each sequence by using the pattern in the differences between terms. 24, 23, 21, 17, 9, 7, ... -7, -39 0, 1, 4, 13, 40, ... 121, 364 7. A triangular number can be represented by dots that are arranged in the shape of an equilateral triangle. The first four triangular numbers are: t₁ = 1, t₂ = 3, t₃ = 6, t₄ = 10.   a. Find the next two triangular numbers. 15, 21 b. Find the tenth triangular number without drawing a diagram. t₁₀ = 55 8. A diagonal of a polygon is a line segment joining any two nonadjacent vertices of the polygon. If n is the number of sides of a polygon (n ≥ 3), then let tₙ be the number of its diagonals.   a. Draw a hexagon and find t₆. t₆ = 9 b. Find t₁₀ without drawing a decagon. t₁₀ = 35 Write a rule for the nth term of the arithmetic sequence. 4, 7, 10, 13, ... tₙ = 4 + (n-1)3 tₙ = 3n + 1 t₅₀ = -444, d = -9 -444 = a₁ + (50-1)(-9) -444 = a₁ - 441 a₁ = -3 tₙ = -3 + (n-1)(-1) tₙ = -n + 2 t₆ = 22, t₁₀ = 38 22 = a₁ + 5d 38 = a₁ + 9d d = 4 a₁ = -2 tₙ = 4n - 2 Find the specified term of each arithmetic sequence. 12. 1, 1.25, 1.5, 1.75, ... t₃₃ t₃₃ = 1 + (32)(0.25) t₃₃ = 9 13. t₁₀ = -29, d = -3; t₄₁ = ? -29 = t₁ + 9(-3) -2 = t₁ t₄₁ = -2 + (40)(-3) t₄₁ = -122 14. t₁₀ = 70, t₁₅ = 60; t₅ = ? 60 = t₁ + 14d 70 = t₁ + 9d -10 = 5d d = -2 60 = t₁ + 14(-2) t₁ = 88 t₅ = 88 + 4(-2) t₅ = 80 Insert a) two and b) three arithmetic means between each pair of numbers. 15. -1, 5 a) d = 2 -1, 1, 3, 5 b) d = 3/2 -1, 1/2, 2, 7/2, 5 16. 14, 39 a) d = 25/3 14, 81/3, 42/3, 39 b) d = 25/4 14, 81/4, 53/2, 31/4, 39 Write a rule for the nth term of the geometric sequence. 17. 7, 14, 28, 56, ... tₙ = 7(2)^(n-1) 18. t₁₂ = 512, r = -2 512 = t₁(2)^11 t₁ = -1/4 tₙ = -1/4(-2)^(n-1) 19. t₂ = 0.1, t₅ = 0.0001 0.0001 = t₁(r)^4 0.1 = t₁(r) r = 1/10 t₁ = 1 tₙ = 1(1/10)^(n-1) Find the specified term of each geometric sequence. 20. 100, 60, 36, ... t₁₆ t₁₆ = 100(3/5)^(16-1) t₁₆ ≈ 0.0470184985 21. t₁ = 16, r = 1/3; t₁₂ t₁₂ = 16(1/3)^(12-1) t₁₂ ≈ 0.00019683 22. t₃ = 12, t₆ = -96; t₁₅ -96 = t₁(r)^5 12 = t₁(r)² r = -2 t₁ = 3 t₁₅ = 3(-2)^14 t₁₅ = 49152 Insert three geometric means between each pair of numbers. -5, -405 -5 * r⁴ = -405 r⁴ = 81 r = ±3 -5, -15, -45, -135, -405 or -5, 15, -45, 135, -405 1/36, 9/4 9/4 = 1/36 * r⁴ 81 = r⁴ r = ±3 1/36, 1/12, 1/4, 3/4, 9/4 or 1/36, -1/12, 1/4, -3/4, 9/4 The number of different electronic games in the arcade seems to have doubled each month since the first one was introduced a year ago. If this description is accurate, how many different games should be in the arcade today? tₙ = 1(2)^(n-1) t₁₂ = 1(2)^11 t₁₂ = 2048 games   If Aunt Sherry sent you $100 on your first birthday, and then increased each successive birthday present by $10, how much will you receive on your twenty-first birthday? tₙ = 100 + (n-1)10 t₂₁ = 100 + 20(10) t₂₁ = 300   An object with negligible air resistance is dropped from an airplane. During the first second of fall, the object falls 4.9 meters; during the second seconds, it falls 14.7 meters; during the third second, it falls 24.5 meters. If this pattern continues, how many meters will the object fall in 10 seconds? d = 9.8 t₁₀ = 4.9 + (10-1)9.8 t₁₀ = 93.1 m   The temperature of water in an ice cube tray is 70°F when it is placed in the freezer. Its temperature n hours after being placed in the freezer is 20% less than 1 hour earlier. Find the temperature of the water a) 6 hours and b) 12 hours after it is placed in the freezer. tₙ = 70(0.8)^n a) t₆ = 70(0.8)^6 ≈ 18.36°F b) t₁₂ = 70(0.8)^12 ≈ 4.8°F   Expand, do not need to find the sum. Σ(n²+7) from n = 1 to 6 8 + 11 + 18 + 27 + 38 + 49 Σ(10-4n) from n = 2 to 6 2 - 2 - 6 - 10 - 14 Σ3(5)^n from n = 1 to 4 15 + 75 + 375 + 1875