Track / Stage 3 / 85 of 260 #85 of 1964

Problem 85

AMC 10/12, early questions

Algebra Difficulty 3.2 Find the answer

In order to draw a graph of $ax^2+bx+c$ , a table of values was constructed. These values of the function for a set of equally spaced increasing values of $x$ were $3844, 3969, 4096, 4227, 4356, 4489, 4624$ , and $4761$ . The one which is incorrect is:
$\text{(A) } 4096$ $\text{(B) } 4356$ $\text{(C) } 4489$ $\text{(D) } 4761$ $\text{(E) } \text{none of these}$

Your answer

Multiple choice: answer with the letter of the option you want.

Official solution

Since the polynomial is quadratic, its second differences must be constant. Taking the first differences, we have
$\begin{align*} 3969-3844&=125 \\ 4096-3969&=127 \\ 4227-4096&=131 \\ 4356-4227&=129 \\ 4489-4356&=133 \\ 4624-4489&=135 \\ 4761-4624&=137 \end{align*}$
This leads to a common second difference of $2$ , with the only discrepancy around the point $4227$ . Observe that if this point were instead $4225$ , the common second difference would, indeed be $2$ for all data points. Therefore the answer is $4227$ , or $\fbox{E}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.