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 the problems: Problem 1: The sum to infinity of a G.P. is -(1)/(10) and the first term is -(1)/(8). Find the common ratio. Step 1: Write down the formula for the sum to infinity of a Geometric Progression (S_). The formula is S_ = (a)/(1-r), where a is the first term and r is the common ratio. Step 2: Substitute the given values into the formula. Given S_ = -(1)/(10) and a = -(1)/(8). -(1)/(10) = (-1)/(8)1-r Step 3: Solve for r. Multiply both sides by (1-r): -(1)/(10)(1-r) = -(1)/(8) Multiply both sides by -10: 1-r = -(1)/(8) × (-10) 1-r = (10)/(8) 1-r = (5)/(4) Subtract 1 from both sides: -r = (5)/(4) - 1 -r = (5)/(4) - (4)/(4) -r = (1)/(4) Multiply by -1: r = -(1)/(4) r = -(1)/(4) --- Problem 2: The fifth term of a G.P. is (2)/(81) if the first term is 2. Find the (i) common ratio and (ii) sum of the first 5 terms of the G.P. i) Find the common ratio (r). Step 1: Write down the formula for the n-th term of a Geometric Progression (a_n). The formula is a_n = ar^n-1, where a is the first term and r is the common ratio. Step 2: Substitute the given values into the formula. Given a_5 = (2)/(81), a = 2, and n=5. (2)/(81) = 2 · r^5-1 (2)/(81) = 2r^4 Step 3: Solve for r. Divide both sides by 2: (2)/(81 × 2) = r^4 (1)/(81) = r^4 Take the fourth root of both sides: r = 4/(81) r = (1)/(3) or r = -(1)/(3) Since the problem does not specify if the terms are positive or negative, both are mathematically possible. However, typically for such problems, the positive root is assumed unless otherwise stated. Let's assume r = (1)/(3) for the sum calculation. If the problem intended a specific sign, it would usually be mentioned. r = (1)/(3) or r = -(1)/(3) ii) Find the sum of the first 5 terms (S_5). Step 1: Write down the formula for the sum of the first n terms of a Geometric Progression (S_n). The formula is S_n = (a(1-r^n))/(1-r) (for r ≠ 1). Step 2: Substitute the values for a, r, and n. Given a = 2, n = 5. Let's use r = (1)/(3). S_5 = (2(1 - (1)/(3))^5)1 - (1)/(3) Step 3: Calculate S_5. First, calculate ((1)/(3))^5: ((1)/(3))^5 = (1^5)/(3^5) = (1)/(243) Now substitute this back into the sum formula: S_5 = (2(1 - 1)/(243))1 - (1)/(3) S_5 = (2(243)/(243) - (1)/(243))(3)/(3) - (1)/(3) S_5 = (2(242)/(243))(2)/(3) S_5 = (484)/(243)(2)/(3) To divide by a fraction, multiply by its reciprocal: S_5 = (484)/(243) × (3)/(2) S_5 = (484 × 3)/(243 × 2) S_5 = (242 × 3)/(243) S_5 = (242)/(81) If we used r = -(1)/(3): S_5 = (2(1 - (-1)/(3))^5)1 - (-(1)/(3)) S_5 = (2(1 - (-1)/(243)))1 + (1)/(3) S_5 = (2(1 + 1)/(243))(4)/(3) S_5 = (2(244)/(243))(4)/(3) S_5 = (488)/(243)(4)/(3) S_5 = (488)/(243) × (3)/(4) S_5 = (122 × 3)/(243) S_5 = (122)/(81) Given the context of typical problems, the positive common ratio is usually intended. S_5 = (242)/(81) What's next?