<p>Write the first five terms of the following sequence and obtain the corresponding series.<br>\(a_1 = a_2 = 2,\ a_n = a_{n-1} - 1,\ n > 2\)</p>
Step-by-Step Solution
Key Concept: A sequence is defined recursively with initial conditions (a₁ = a₂ = 2) and a recurrence relation (aₙ = aₙ₋₁ - 1 for n > 2). Generate terms by successively applying the recurrence relation, then write the corresponding series as the sum of these terms.
<p><strong>Step 1:</strong> Identify the given conditions.</p><p>Initial terms: a₁ = 2, a₂ = 2</p><p>Recurrence relation: aₙ = aₙ₋₁ - 1 for n > 2</p><p><strong>Step 2:</strong> Generate the first five terms by applying the recurrence relation starting from n = 3.</p><p>a₃ = a₂ - 1 = 2 - 1 = 1</p><p>a₄ = a₃ - 1 = 1 - 1 = 0</p><p>a₅ = a₄ - 1 = 0 - 1 = -1</p><p><strong>Step 3:</strong> Write the first five terms.</p><p>The sequence is: <strong>2, 2, 1, 0, -1</strong></p><p><strong>Step 4:</strong> Write the corresponding series.</p><p>The series is the sum: 2 + 2 + 1 + 0 + (-1) = 4 or ∑(aₙ) from n=1 to 5 equals 4</p><p>∴ Answer: 2, 2, 1, 0, -1</p>
Correct Answer: 2, 2, 1, 0, -1