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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Question 1: a) Find the first 5 terms of the sequence defined by a₁ = 3 and aₙ = 2aₙ₋₁ + 1 for n ≥ 2. Step 1: Use the given recursive formula to find each term. a₁ = 3 (given) a₂ = 2a₁ + 1 = 2(3) + 1 = 6 + 1 = 7 a₃ = 2a₂ + 1 = 2(7) + 1 = 14 + 1 = 15 a₄ = 2a₃ + 1 = 2(15) + 1 = 30 + 1 = 31 a₅ = 2a₄ + 1 = 2(31) + 1 = 62 + 1 = 63 The first 5 terms are 3, 7, 15, 31, 63. b) Is this sequence arithmetic or geometric? Step 1: Check for a common difference (arithmetic). 7 - 3 = 4 15 - 7 = 8 Since the difference is not constant, it is not an arithmetic sequence. Step 2: Check for a common ratio (geometric). 7 / 3 ≈ 2.33 15 / 7 ≈ 2.14 Since the ratio is not constant, it is not a geometric sequence. This sequence is neither arithmetic nor geometric. c) Find a closed-form expression for aₙ. Step 1: Observe the pattern in the terms: 3, 7, 15, 31, 63. These numbers are close to powers of 2: 4, 8, 16, 32, 64. Specifically, each term is 1 less than a power of 2. a₁ = 3 = 2² - 1 a₂ = 7 = 2³ - 1 a₃ = 15 = 2⁴ - 1 a₄ = 31 = 2⁵ - 1 a₅ = 63 = 2⁶ - 1 Step 2: Identify the relationship between the term number (n) and the exponent. The exponent is always n + 1. Step 3: Write the closed-form expression. The closed-form expression is aₙ = 2^(n+1) - 1. Send me the next one 📸