How do you express a linear sequence derived from a quadratic sequence?

To express a linear sequence derived from a quadratic sequence, one effective method is to consider the differences between the terms of the quadratic sequence. Quadratic sequences have a second difference that is constant; when you take the first differences of the terms of the sequence, you will find a linear sequence.

To clarify the reasoning behind the selected approach: if you have a quadratic sequence, denoted by (a_n = An^2 + Bn + C), taking the terms of the sequence and subtracting a function that relates to the square of the term positions (like (n^2)) can yield a linear representation of the differences between terms. This is often described in terms of the (a_n - a_{n-1}^2) or a similar formulation, effectively isolating the linear portion of the sequence.

This method allows you to reduce the quadratic complexity down to a linear one. In essence, the selected approach succinctly captures the transformation of a quadratic relationship into a linear sequence, clarifying the underlying structure of the terms.

The other options do not accurately originate a linear sequence from a quadratic one. Taking the square root would not simplify the sequence appropriately, adding a constant term does not yield the differences relevant to