Quadtratic Patterns Question

Question

Consider the sequence below:

11, 14, 19, 26, 35 …

What explicit expression can be used to find the nth term of this sequence?

Answer 1

A sequence is quadratic if its terms have the explicit form a_n = An² + Bn + C. The first three terms of the given sequence are a₁ = 11, a₂ = 14, and a₃ = 19. Substituting these into the explicit form gives a system of three equations in three variables:

11 = A + B + C

14 = 4A + 2B + C

19 = 9A + 3B + C

Solve this system to get A = 1, B = 0, C = 10, which means that a_n = n² + 10.

Answer 2

11, 11+3, 14+5, 19+7, 26+9

a_n = 11 + (2n + 1)