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The characteristic equation is a powerful tool in solving linear recurrence relations like those found in the Pell sequence. When dealing with a recurrence relation, we often assume solutions of the form \(P_n = r^n\). By substituting this assumed form into the recurrence equation \(P_{n+2} = 2P_{n+1} + P_n\), we can derive a polynomial equation known as the characteristic equation. For our specific case, this substitution leads to the equation \(r^{n+2} = 2r^{n+1} + r^n\). Dividing each term by \(r^n\), we simplify this to \(r^2 = 2r + 1\), resulting in the quadratic characteristic equation \(r^2 - 2r - 1 = 0\). This equation will allow us to determine the roots, which in turn will help in formulating a general solution for the sequence.

A recurrence relation defines each term of a sequence using preceding terms. For the Pell sequence, this is given by the formula \(P_{n+2} = 2P_{n+1} + P_n\). It essentially prescribes how each term is generated from the previous two terms, setting into motion a predictable pattern.The relation starts with initial conditions, in this case, \(P_1 = 1\) and \(P_2 = 2\). As one computes further terms, these initial conditions serve as the foundation. Recurrence relations are important in mathematical sequences as they provide a structured way to understand and compute terms without having to explicitly outline every term in the sequence. Through the recurrence relation, a larger pattern or formula can often be derived, simplifying computation and analysis.

The quadratic formula is crucial for solving the characteristic equation derived from our recurrence relation. Given a standard quadratic equation of the form \(ax^2 + bx + c = 0\), the quadratic formula enables us to find its roots as follows:\[\x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\]For our characteristic equation \(r^2 - 2r - 1 = 0\), we plug in \(a = 1\), \(b = -2\), and \(c = -1\). First, we compute the discriminant, \(b^2 - 4ac\), which equals 8. Then we insert these values into the quadratic formula, providing us with the roots: \(r = 1 + \sqrt{2}\) and \(r = 1 - \sqrt{2}\). These roots play an essential role in constructing the general solution for the sequence.

The general solution of a recurrence relation describes the overall pattern of the sequence based on the roots found from the characteristic equation. For the Pell sequence, the roots \(r_1 = 1 + \sqrt{2}\) and \(r_2 = 1 - \sqrt{2}\) yield a general solution of the form:\[\P_n = A(1+\sqrt{2})^n + B(1-\sqrt{2})^n\\]To determine the constants \(A\) and \(B\), we must use the initial conditions \(P_1 = 1\) and \(P_2 = 2\). Solving this system of equations gives us \(A = \frac{1}{2\sqrt{2}}\) and \(B = -\frac{1}{2\sqrt{2}}\). Hence, the full explicit formula for the Pell sequence becomes:\[\P_n = \frac{1}{2\sqrt{2}} [(1 + \sqrt{2})^n - (1 - \sqrt{2})^n]\\]This representation provides a closed formula, allowing us to compute terms of the sequence directly without recursive calculations.