This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — The formula for the sum of the first n terms of a geometric series is S_n = (a(r^n - 1))/(r-1) or S_n = (a(1 - r^n))/(1-r), where a is the first term and r is the common ratio. 1.a) Step 1: Identify the first term (a), common ratio (r), and number of terms (n). The series is (2)/(27) + (2)/(9) + (2)/(3) + to 9 terms. a = (2)/(27) r = (2/9)/(2/27) = (2)/(9) × (27)/(2) = 3 n = 9 Step 2: Apply the sum formula. S_9 = (a(r^n - 1))/(r-1) S_9 = (2)/(27)(3^9 - 1)3-1 S_9 = (2)/(27)(19683 - 1)2 S_9 = (2)/(27)(19682)2 S_9 = (19682)/(27) The sum of the series is (19682)/(27). 1.b) Step 1: Identify the first term (a), common ratio (r), and number of terms (n). The series is -64 + 32 - 16 + to 10 terms. a = -64 r = (32)/(-64) = -(1)/(2) n = 10 Step 2: Apply the sum formula. Since |r| < 1, we use S_n = (a(1 - r^n))/(1-r). S_10 = (-64(1 - (-1)/(2))^10)1 - (-(1)/(2)) S_10 = (-64(1 - 1)/(1024))1 + (1)/(2) S_10 = (-64(1024 - 1)/(1024))(3)/(2) S_10 = (-64(1023)/(1024))(3)/(2) S_10 = -64 × (1023)/(1024) × (2)/(3) S_10 = -(1023 × 2)/(16 × 3) S_10 = -(1023)/(8 × 3) S_10 = -(341)/(8) The sum of the series is -(341)/(8). 2) Step 1: Identify the first term (a), common ratio (r), and number of terms (n). The sequence is (1)/(4), -(1)/(2), 1, and we need the sum of the first 12 terms. a = (1)/(4) r = (-1/2)/(1/4) = -(1)/(2) × 4 = -2 n = 12 Step 2: Apply the sum formula. S_12 = (a(r^n - 1))/(r-1) S_12 = (1)/(4)((-2)^12 - 1)-2-1 S_12 = (1)/(4)(4096 - 1)-3 S_12 = (1)/(4)(4095)-3 S_12 = (4095)/(-12) S_12 = -(1365)/(4) The sum of the first 12 terms is -(1365)/(4). 3.a) Step 1: Identify the first term (a), common ratio (r), and number of terms (n) from the summation. The summation is _K=1^10 (1)/(50)(5)^K-1. This is a geometric series. For K=1, the first term a = (1)/(50)(5)^1-1 = (1)/(50)(5)^0 = (1)/(50) × 1 = (1)/(50). The common ratio r = 5. The number of terms n = 10 - 1 + 1 = 10. Step 2: Apply the sum formula. S_10 = (a(r^n - 1))/(r-1) S_10 = (1)/(50)(5^10 - 1)5-1 S_10 = (1)/(50)(5^10 - 1)4 S_10 = 5^10 - 1200 S_10 = (9765625 - 1)/(200) S_10 = (9765624)/(200) S_10 = 48828.12 The sum is 48828.12. 3.b) Step 1: Identify the first term (a), common ratio (r), and number of terms (n) from the summation. The summation is _i=0^10 3^4-i. This is a geometric series. For i=0, the first term a = 3^4-0 = 3^4 = 81. For i=1, the second term is 3^4-1 = 3^3 = 27. The common ratio r = (27)/(81) = (1)/(3). The number of terms n = 10 - 0 + 1 = 11. Step 2: Apply the sum formula. Since |r| < 1, we use S_n = (a(1 - r^n))/(1-r). S_11 = (81(1 - (1)/(3))^11)1 - (1)/(3) S_11 = (81(1 - 1)/(3^11))(2)/(3) S_11 = 3^4(3^11 - 13^11\