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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Here's the solution to question 8: The nth term of a geometric progression (G.P.) is given by T_n = 5 × 2^2-n. a) Write down the first four terms of the G.P. Step 1: Calculate the first term (n=1). T_1 = 5 × 2^2-1 = 5 × 2^1 = 5 × 2 = 10 Step 2: Calculate the second term (n=2). T_2 = 5 × 2^2-2 = 5 × 2^0 = 5 × 1 = 5 Step 3: Calculate the third term (n=3). T_3 = 5 × 2^2-3 = 5 × 2^-1 = 5 × (1)/(2) = (5)/(2) Step 4: Calculate the fourth term (n=4). T_4 = 5 × 2^2-4 = 5 × 2^-2 = 5 × (1)/(4) = (5)/(4) The first four terms are: 10, 5, (5)/(2), (5)/(4) b) Calculate the sum of the first six terms of this series. Step 1: Identify the first term (a) and the common ratio (r). From part (a), the first term is a = T_1 = 10. The common ratio r is found by dividing any term by its preceding term: r = (T_2)/(T_1) = (5)/(10) = (1)/(2) Step 2: Use the formula for the sum of the first n terms of a G.P. The formula for the sum of the first n terms is S_n = (a(1 - r^n))/(1 - r). We need to find the sum of the first six terms, so n=6, a=10, and r=(1)/(2). S_6 = (10 (1 - (1)/(2))^6)1 - (1)/(2) Step 3: Calculate (1/2)^6. ((1)/(2))^6 = (1^6)/(2^6) = (1)/(64) Step 4: Substitute the value and simplify. S_6 = (10 (1 - 1)/(64))(1)/(2) S_6 = (10 (64)/(64) - (1)/(64))(1)/(2) S_6 = (10 (63)/(64))(1)/(2) S_6 = 10 × (63)/(64) × 2 S_6 = (20 × 63)/(64) S_6 = (5 × 63)/(16) S_6 = (315)/(16) The sum of the first six terms is: (315)/(16)