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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.

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Step-by-step solution

Part 1

Part a· 1 step

Use the given recursive formula to find each term.

Part b· 2 steps

Check for a common difference (arithmetic).

Check for a common ratio (geometric).

Part c· 3 steps

Observe the pattern in the terms

Identify the relationship between the term number (n) and the exponent.

Write the closed-form expression.

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