To find the nth term of a quadratic sequence, what value should you divide the second difference by?

For a quadratic sequence, the nth term can be expressed in the form of a quadratic equation, typically written as ( an^2 + bn + c ). To derive this expression, we examine the differences between terms of the sequence.

First, we find the first differences, which are the differences between consecutive terms of the sequence. If we continue to find the differences of these first differences, we arrive at the second differences.

In the case of a quadratic sequence, the second differences are constant. This constancy arises from the nature of quadratic functions. To determine the coefficient ( a ) in the quadratic expression, we take the second difference and divide it by 2. This is due to how quadratic equations evolve; they produce curvature that results in a constant second difference.

Therefore, to find ( a ), dividing the constant second difference by 2 gives the leading coefficient of the quadratic term. Hence, in the context of the question, the correct choice is to divide the second difference by 2. This reflects the fundamental properties of quadratic functions in sequences.