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.
ScanSolve AI Answer
Dalu💕, let's knock this out. To find the 7th term (a₇) of the sequence, we use the given recurrence relation aₙ₊₂ = 2aₙ₊₁ - aₙ + 3 and the initial terms a₁ = 1 and a₂ = 2. We will calculate each term step-by-step. Step 1: Identify the given terms and the recurrence relation. The first two terms are: a₁ = 1 a₂ = 2 The recurrence relation is: aₙ₊₂ = 2aₙ₊₁ - aₙ + 3 Step 2: Calculate a₃. Substitute n = 1 into the recurrence relation: a₁₊₂ = 2a₁₊₁ - a₁ + 3 a₃ = 2a₂ - a₁ + 3 a₃ = 2(2) - 1 + 3 a₃ = 4 - 1 + 3 a₃ = 6 Step 3: Calculate a₄. Substitute n = 2 into the recurrence relation: a₂₊₂ = 2a₂₊₁ - a₂ + 3 a₄ = 2a₃ - a₂ + 3 a₄ = 2(6) - 2 + 3 a₄ = 12 - 2 + 3 a₄ = 13 Step 4: Calculate a₅. Substitute n = 3 into the recurrence relation: a₃₊₂ = 2a₃₊₁ - a₃ + 3 a₅ = 2a₄ - a₃ + 3 a₅ = 2(13) - 6 + 3 a₅ = 26 - 6 + 3 a₅ = 23 Step 5: Calculate a₆. Substitute n = 4 into the recurrence relation: a₄₊₂ = 2a₄₊₁ - a₄ + 3 a₆ = 2a₅ - a₄ + 3 a₆ = 2(23) - 13 + 3 a₆ = 46 - 13 + 3 a₆ = 36 Step 6: Calculate a₇. Substitute n = 5 into the recurrence relation: a₅₊₂ = 2a₅₊₁ - a₅ + 3 a₇ = 2a₆ - a₅ + 3 a₇ = 2(36) - 23 + 3 a₇ = 72 - 23 + 3 a₇ = 52 The sequence of terms is 1, 2, 6, 13, 23, 36, 52. The method shown in your notes, using a difference table, also confirms this result. The first differences are 1, 4, 7, 10, 13, 16. The second differences are constant at 3, which indicates a quadratic sequence. The 7th term, a₇, is 52. Send me the next one 📸