Which general term is correct for the sequence (3, 18, 43, 78, ...)?

Correct answer: aₙ = 5n² − 2

The governing concept is recognizing a quadratic sequence from its second differences and verifying its explicit rule. The first differences are 18 − 3 = 15, 43 − 18 = 25, and 78 − 43 = 35. The second differences are 25 − 15 = 10 and 35 − 25 = 10, so a quadratic rule is appropriate. Test option A: when n = 1, 5(1)² − 2 = 3; n = 2 gives 20 − 2 = 18; n = 3 gives 45 − 2 = 43; and n = 4 gives 80 − 2 = 78. Therefore option A is correct. Option B is linear and gives 3, 8, 13, ...; option C gives 3, 12, 27, ...; and option D gives 3, 18, 33, ..., so none reproduces all the terms.