Model Answer:

Step 1: Check if the sequence is arithmetic.

Differences: 8 - 5 = 3, 12 - 8 = 4, 18 - 12 = 6. The differences are not constant, so it’s not arithmetic.

Step 2: Look for a quadratic pattern by examining second differences.

First differences: 3, 4, 6. Second differences: 4 - 3 = 1, 6 - 4 = 2. Inconsistent, so check for a quadratic nth term.

Step 3: Assume nth term is an² + bn + c. Use the first three terms.

n = 1: a(1)² + b(1) + c = a + b + c = 5 (1).

n = 2: a(2)² + b(2) + c = 4a + 2b + c = 8 (2).

n = 3: a(3)² + b(3) + c = 9a + 3b + c = 12 (3).

Solve: Subtract (1) from (2): 3a + b = 3 (4).

Subtract (2) from (3): 5a + b = 4 (5).

Subtract (4) from (5): 2a = 1 → a = 0.5.

Substitute a = 0.5 into (4): 3(0.5) + b = 3 → 1.5 + b = 3 → b = 1.5.

Substitute a = 0.5, b = 1.5 into (1): 0.5 + 1.5 + c = 5 → c = 3.

Step 4: Write the nth term.

nth term = 0.5n² + 1.5n + 3.

Final Answer: 0.5n² + 1.5n + 3

Nancy's Non Calculator Maths Revision