A-Level Sequences and series: worked solution

Question

A sequence has nth term u_n = a n² + b n + c. Its opening three terms are 2, 6 and 12. Work out a, b and c. [3]

The nth term is u_n = a n² + b n + c, so substitute n = 1, 2, 3 in turn and set each equal to the given term. n = 1: a(1)² + b(1) + c = a + b + c = 2 ... (1) n = 2: a(2)² + b(2) + c = 4a + 2b + c = 6 ... (2) n = 3: a(3)² + b(3) + c = 9a + 3b + c = 12 ... (3) Every equation contains the same +c, so subtracting consecutive equations eliminates c. (2) - (1): (4a - a) + (2b - b) = 6 - 2, so 3a + b = 4 ... (4) (3) - (2): (9a - 4a) + (3b - 2b) = 12 - 6, so 5a + b = 6 ... (5) Now both (4) and (5) contain the same +b, so subtract again to eliminate b. (5) - (4): 2a = 2, so a = 1 Substitute a = 1 into (4): 3(1) + b = 4, so b = 1 Substitute a = 1 and b = 1 into (1): 1 + 1 + c = 2, so c = 0 Check: u_n = n² + n gives u_1 = 1 + 1 = 2, u_2 = 4 + 2 = 6, u_3 = 9 + 3 = 12. ✓ So a = 1, b = 1, c = 0.