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 are the solutions to your math problems. 1. Step 1: Set up equations for the given terms of the Geometric Progression (G.P.). The n-th term of a G.P. is given by T_n = ar^n-1, where a is the first term and r is the common ratio. Given: Second term (T_2) = 8 Fourth term (T_4) = 32 From the formula: T_2 = ar^2-1 = ar = 8 (Equation 1) T_4 = ar^4-1 = ar^3 = 32 (Equation 2) Step 2: Find the common ratio (r). Divide Equation 2 by Equation 1: (ar^3)/(ar) = (32)/(8) r^2 = 4 r = sqrt(4) Since the terms are positive, we take the positive root: r = 2 Step 3: Find the first term (a). Substitute r=2 into Equation 1: a(2) = 8 a = (8)/(2) a = 4 Step 4: Calculate the sum of the first nine terms (S_9). The sum of the first n terms of a G.P. is given by S_n = (a(r^n - 1))/(r - 1) for r ≠ 1. Here, a=4, r=2, and n=9. S_9 = (4(2^9 - 1))/(2 - 1) S_9 = (4(512 - 1))/(1) S_9 = 4(511) S_9 = 2044 The sum of the first nine terms is 2044. 2. Step 1: Set up equations for the given terms of the G.P. Given: Third term (T_3) = 48 Seventh term (T_7) = (3)/(4) From the formula T_n = ar^n-1: T_3 = ar^3-1 = ar^2 = 48 (Equation 1) T_7 = ar^7-1 = ar^6 = (3)/(4) (Equation 2) Step 2: Find the common ratio (r). Divide Equation 2 by Equation 1: (ar^6)/(ar^2) = (3/4)/(48) r^4 = (3)/(4 × 48) r^4 = (3)/(192) r^4 = (1)/(64) Step 3: Find the product of the 2nd and 4th terms. The 2nd term is T_2 = ar. The 4th term is T_4 = ar^3. The product of the 2nd and 4th terms is T_2 × T_4 = (ar)(ar^3) = a^2 r^4. From Equation 1, we have ar^2 = 48. We can square this to get a^2 r^4: (ar^2)^2 = 48^2 a^2 r^4 = 2304 Alternatively, substitute r^4 = (1)/(64) into a^2 r^4. First, find a: From ar^2 = 48 and r^4 = (1)/(64), we know r^2 = sqrt((1)/(64)) = (1)/(8) (assuming r^2 is positive). a ((1)/(8)) = 48 a = 48 × 8 = 384 Now, calculate a^2 r^4: a^2 r^4 = (384)^2 × (1)/(64) a^2 r^4 = 147456 × (1)/(64) a^2 r^4 = 2304 The product of its 2nd and 4th terms is 2304. Drop the next question.