Recursive rule

Direct answer: a₅ = 45, so option A is correct. The recurrence tells us how to move from aₙ to aₙ₊₁: subtract 7n/2. Since the starting term is a₁, we must use n = 1, 2, 3, and 4 to reach a₅. Step by step, a₂ = 80 − 7(1)/2 = 76.5; a₃ = 76.5 − 7(2)/2 = 69.5; a₄ = 69.5 − 7(3)/2 = 59; and a₅ = 59 − 7(4)/2 = 45. A quicker check combines the subtractions: (7/2)(1 + 2 + 3 + 4) = (7/2)(10) = 35, so a₅ = 80 − 35 = 45. Option A matches both methods. Option B, 48, leaves too little total subtraction. Option C, 51, usually comes from omitting one part of the fractional decrease. Option D, 54, also uses an incorrect total decrease. Warning: reaching a₅ requires four transitions, not five, because a₁ is already given.

The governing concept is a recursive sequence whose increment depends on the index n. Start with a₁ = 5 and apply the rule aₙ₊₁ = aₙ + n² one step at a time. For n = 1, a₂ = 5 + 1² = 6. For n = 2, a₃ = 6 + 2² = 10. For n = 3, a₄ = 10 + 3² = 19. For n = 4, a₅ = 19 + 4² = 35. For n = 5, a₆ = 35 + 5² = 60. Therefore 60 is the sixth term, so option A is correct. The increments are 1, 4, 9, 16, and 25, the consecutive squares required by the recurrence. The seventh, eighth, and ninth terms come after a₆ and are larger than 60, so options B, C, and D cannot be the first position of 60.

A recursive rule must reproduce later terms from the stated starting terms. Test option D using a₁ = 3 and a₂ = 8. For n = 3, a₃ = 2a₂ + a₁ = 2(8) + 3 = 19. For n = 4, a₄ = 2a₃ + a₂ = 2(19) + 8 = 46. Thus the rule generates both given terms and option D is correct. Option A gives a₃ = 8 + 2(3) = 14, so it fails immediately. Option B uses the wrong initial values, even though its expression resembles the required pattern. Option C produces a₃ = 8 + 3 + 8 = 19, but then produces a₄ = 19 + 8 + 8 = 35, not 46. Testing more than one step ensures the rule is genuinely correct.