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.
ScanSolve AI Answer
Okay Austino, let's tackle these. 17. Step 1: Substitute n=6 into the given formula for S_n. The formula is S_n = 2n^2 - 3n. We need to find S_6. S_6 = 2(6)^2 - 3(6) Step 2: Calculate the value. S_6 = 2(36) - 18 S_6 = 72 - 18 S_6 = 54 The value of S_6 is 54. The correct option is B. B. 54 18. Step 1: Define variables and set up the probability equation. Let the number of green balls be G = 20. Let the number of red balls be R. The total number of balls is T = G + R = 20 + R. The probability of selecting a red ball is P(red) = Number of red ballsTotal number of balls = (R)/(20+R). Given that P(red) = (3)/(5). So, we have the equation: (R)/(20+R) = (3)/(5) Step 2: Solve the equation for R. Cross-multiply: 5R = 3(20+R) 5R = 60 + 3R Subtract 3R from both sides: 5R - 3R = 60 2R = 60 Divide by 2: R = (60)/(2) R = 30 There are 30 red balls. Step 3: Calculate the total number of balls in the bag. Total balls T = 20 + R = 20 + 30. T = 50 There are 50 balls in the bag. The correct option is B. B. 50 19. Step 1: Write expressions for the terms of the Arithmetic Progression. The first term is a = 20. The n-th term of an AP is given by a_n = a + (n-1)d, where d is the common difference. The third term is a_3 = a + (3-1)d = a + 2d. The fifth term is a_5 = a + (5-1)d = a + 4d. Step 2: Use the given condition to form an equation. The fifth term is four-fifth of the third term: a_5 = (4)/(5) a_3 Step 3: Substitute the expressions for a_3 and a_5 into the equation. a + 4d = (4)/(5)(a + 2d) Step 4: Substitute a=20 and solve for d. 20 + 4d = (4)/(5)(20 + 2d) Multiply both sides by 5 to eliminate the fraction: 5(20 + 4d) = 4(20 + 2d) 100 + 20d = 80 + 8d Subtract 8d from both sides: 100 + 20d - 8d = 80 100 + 12d = 80 Subtract 100 from both sides: 12d = 80 - 100 12d = -20 Divide by 12: d = -(20)/(12) Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 4: d = -(20 ÷ 4)/(12 ÷ 4) d = -(5)/(3) The common difference is -(5)/(3). The correct option is C. C. -(5)/(3) That's 2 down. 3 left today — send the next one.