Identify the first term (a) and the common ratio (r).

Here are the solutions to the questions: 4. Find the ninth term and the general term of the progression (1)/(4), -(1)/(2), 1, -2. Step 1: Identify the first term (a) and the common ratio (r). The first term is a = (1)/(4). The common ratio r is found by dividing any term by its preceding term: r = (-1)/(2)(1)/(4) = -(1)/(2) × 4 = -2. Check with the next terms: (1)/(-1)2 = -2, (-2)/(1) = -2. This is a geometric progression (GP). Step 2: Find the general term (T_n). The formula for the general term of a GP is T_n = a · r^n-1. Substitute a = (1)/(4) and r = -2: T_n = (1)/(4) (-2)^n-1 Step 3: Find the ninth term (T_9). Substitute n=9 into the general term formula: T_9 = (1)/(4) (-2)^9-1 T_9 = (1)/(4) (-2)^8 Calculate (-2)^8: (-2)^8 = 256 T_9 = (1)/(4) × 256 T_9 = 64 The general term is T_n = (1)/(4)(-2)^n-1 and the ninth term is 64. 5. Find the sum of the first 8 terms of the G.P with a=5 and r=2. Step 1: Use the formula for the sum of the first n terms of a GP. The formula for the sum of the first n terms of a GP when r ≠ 1 is S_n = (a(r^n - 1))/(r - 1). Given a=5, r=2, and n=8. Step 2: Substitute the values into the formula. S_8 = (5(2^8 - 1))/(2 - 1) S_8 = (5(256 - 1))/(1) S_8 = 5(255) S_8 = 1275 The sum of the first 8 terms is 1275. 6. The 1^st and 6^th terms of a G.P are 153 and 17/27. Find the sum of the first four terms. Step 1: Write down the given information using the GP term formula T_n = a · r^n-1. T_1 = a = 153 T_6 = a · r^6-1 = a · r^5 = (17)/(27) Step 2: Find the common ratio (r). Substitute a=153 into the equation for T_6: 153 · r^5 = (17)/(27) r^5 = (17)/(27 × 153) Simplify the fraction: 153 = 9 × 17. r^5 = (17)/(27 × 9 × 17) r^5 = (1)/(27 × 9) r^5 = (1)/(3^3 × 3^2) r^5 = (1)/(3^5) Therefore, r = (1)/(3). Step 3: Find the sum of the first four terms (S_4). The formula for the sum of the first n terms of a GP when |r| < 1 is S_n = (a(1 - r^n))/(1 - r). Given a=153, r=(1)/(3), and n=4. S_4 = (153 (1 - (1)/(3))^4)1 - (1)/(3) Calculate ((1)/(3))^4: ((1)/(3))^4 = (1)/(81) Calculate 1 - (1)/(3): 1 - (1)/(3) = (2)/(3) Substitute these values back into the formula: S_4 = (153 (1 - 1)/(81))(2)/(3) S_4 = (153 (81 - 1)/(81))(2)/(3) S_4 = (153 (80)/(81))(2)/(3) S_4 = 153 × (80)/(81) × (3)/(2) Simplify the expression: S_4 = (153 × 80 × 3)/(81 × 2) S_4 = ((9 × 17) × (2 × 40) × 3)/((9 × 9) × 2) S_4 = (17 × 40 × 3)/(9) S_4 = (17 × 40 × 1)/(3) S_4 = (680)/(3) The sum of the first four terms is (680)/(3). 7. If T_2=35 and T_4=875 respectively. Find the first and fifth terms. Step 1: Write down the given information using the GP term formula T_n = a · r^n-1. T_2 = a · r^2-1 = a · r = 35 (Equation 1) T_4 = a · r^4-1 = a · r^3 = 875 (Equation 2) Step 2: Find the common ratio (r). Divide Equation 2 by Equation 1: (a · r^3)/(a · r) = (875)/(35) r^2 = 25 r = ± sqrt(25) r = ± 5 Step 3: Find the first term (a) and the fifth term (T_5) for each possible value of r. Case 1: r=5 Substitute r=5 into Equation 1: a · 5 = 35 a = (35)/(5) a = 7 The first term is T_1 = a = 7. The fifth term is T_5 = a · r^5-1 = a · r^4: T_5 = 7 · (5)^4 T_5 = 7 · 625 T_5 = 4375 Case 2: r=-5