Final answer:
To find a closed form solution for the given sequence, we can use the characteristic equation. The characteristic equation is found by rearranging the given recurrence relation and assuming a closed form solution of the form gn = r^n. The characteristic equation is then solved to find the roots.
Step-by-step explanation:
To find a closed form solution for the coefficients of the sequence (90, 91, 92, ...), we can use the characteristic equation. Let's first determine the characteristic equation. We are given that gn = 79gn-1 + 129gn-2 for all n > 2. Rearranging the equation, we have gn - 79gn-1 - 129gn-2 = 0. The general solution for the sequence is obtained by substituting the roots into a formula, where the constants are determined by the initial conditions.
To find the characteristic equation, we assume that the sequence has a closed form solution of the form gn = r^n for some constant r. Substituting this into the equation and simplifying, we get r^n - 79r^(n-1) - 129r^(n-2) = 0.
Now, we can rewrite the equation in terms of powers of r: r^2 - 79r - 129 = 0. This is the characteristic equation. Solving this quadratic equation, we find the roots r1 = 9 and r2 = -14.
The general solution for the sequence is given by the formula gn = A*r1^n + B*r2^n, where A and B are constants determined by the initial conditions. Since we are given that 90 = 2 and 91 = 1, we can substitute these values into the formula to find the values of A and B. Solving the resulting system of equations, we find A = 13/23 and B = -15/23.
Therefore, the closed form solution for the coefficients of the sequence (90, 91, 92, ...) is given by gn = (13/23)*9^n + (-15/23)*(-14)^n.