Find a quadratic model for the sequence with the indicated terms.

Question

Quizplus · Accepted Answer

The common difference between terms is not constant, so the sequence is not arithmetic. However, the second differences are constant -4, indicating the sequence is quadratic. To find the quadratic equation, we use the formula for finding the coefficients of a quadratic model from three data points, which is
$f(x) = ax^2 + bx + c$, where
$a = \frac{f(c) - f(b)}{(c-b)(c-a)}$, 
$b = \frac{f(c) - f(a)}{(c-a)} - a(c+a)$, and
$c = f(a)$.
Using the 3rd, 6th, and 11th terms, we get:
$a = \frac{a_{11} - a_6}{(11-6)(11-3)} = \frac{127 - 58}{30} = \frac{69}{30} = \frac{23}{10}$
$b = \frac{a_{11} - a_3}{11-3} - a(11+3) = \frac{969-13}{8} - \frac{23}{10}(14) = \frac{463}{5}$
$c = a_3 = -3$
So the quadratic equation is $f(x) = \frac{23}{10}x^2 + \frac{463}{5}x - 3$. The correct choice is B).

Find a Quadratic Model for the Sequence with the Indicated