Solve the recurrence relation \( \mathrm{S}(\mathrm{K})-\mathrm{S}(\mathrm{K}-1)-\mathrm{S}(\mathrm{K}-2)=0 \).

To solve the recurrence relation \[ S(K) - S(K-1) - S(K-2) = 0, \] we can rearrange it to express \( S(K) \) in terms of the previous two terms: \[ S(K) = S(K-1) + S(K-2). \] This is a second-order linear homogeneous recurrence relation. The characteristic equation associated with this recurrence is obtained by assuming a solution of the form \( S(K) = r^K \): \[ r^K - r^{K-1} - r^{K-2} = 0. \] Dividing through by \( r^{K-2} \) (assuming \( r \neq 0 \)), we get: \[ r^2 - r - 1 = 0. \] To find the roots of this characteristic equation, we can use the quadratic formula: \[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{1 \pm \sqrt{1^2 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}. \] Let \( r_1 = \frac{1 + \sqrt{5}}{2} \) and \( r_2 = \frac{1 - \sqrt{5}}{2} \). These roots are the golden ratio \( \phi \) and its conjugate. The general solution to the recurrence relation is given by: \[ S(K) = A r_1^K + B r_2^K, \] where \( A \) and \( B \) are constants determined by the initial conditions. To fully specify the solution, we need initial conditions \( S(0) \) and \( S(1) \). Let's denote these as: - \( S(0) = S_0 \) - \( S(1) = S_1 \) Using these initial conditions, we can set up the following equations: 1. For \( K = 0 \): \[ S(0) = A r_1^0 + B r_2^0 = A + B = S_0. \] 2. For \( K = 1 \): \[ S(1) = A r_1^1 + B r_2^1 = A r_1 + B r_2 = S_1. \] Now we have a system of linear equations: \[ \begin{align*} A + B &= S_0, \\ A r_1 + B r_2 &= S_1. \end{align*} \] We can solve this system for \( A \) and \( B \). From the first equation, we can express \( B \) in terms of \( A \): \[ B = S_0 - A. \] Substituting this into the second equation gives: \[ A r_1 + (S_0 - A) r_2 = S_1. \] Expanding and rearranging: \[ A r_1 + S_0 r_2 - A r_2 = S_1, \] \[ A (r_1 - r_2) = S_1 - S_0 r_2. \] Thus, \[ A = \frac{S_1 - S_0 r_2}{r_1 - r_2}. \] Now substituting \( A \) back into the equation for \( B \): \[ B = S_0 - \frac{S_1 - S_0 r_2}{r_1 - r_2}. \] Now we have \( A \) and \( B \) in terms of the initial conditions \( S_0 \) and \( S_1 \). Finally, the solution to the recurrence relation is: \[ S(K) = A r_1^K + B r_2^K, \] where \( A \) and \( B \) are determined from the initial conditions.