Find the explicit formulas for the following second-order linear homogeneous rec

Question

Find the explicit formulas for the following second-order linear homogeneous recurrence relations a_n = 6a_n - 1 - 8a_n - 2, a_0 = 1, a_1 = 0; 2a_n = 7a_n - 1 - 3a_n - 2, a_0 = 1, a_1 = 1; a_n = a_n -1 + a_n - 2, a_0 = 1, a_1 = 1; a_n = 4(a_n - 1 - a_n - 2), a_0 = 1, a_1 = 1; Find the degree of each vertix for the following graphs Decide whether the graph has an euler cycle. If the graph has an euler cycle, exhibit one. Decide whether the following graph has a Hamiltonian cycle With respect to the following graph, find the length of a shortest path and a shortest path between each pair of vertices in the weighted graph. b, and j a, and f.

Explanation / Answer

1. a> the characteristic equation is given as :

t^2 - 6t + 8 = 0

(t-4)(t-2) = 0

therefore the roots are : r = 4 and s = 2

By the distinct roots theorem :

C+D = a0 = 1 and

Cr + Ds = a1 = 0

So we have Two equations, two unknowns, so we'll solve for the unknowns

=> C+D = 1 and 4C+2D = 0

=> C = -1 and D = 2

Hence the explicit formula is :

an = Cr^n + Ds^n , for n3 0

an = -4^n + 2*2^n

an = -(4)^n + 2^(n+1) , for n3 0

lets check

at n=0

a0 = -(4)^(0) + 2^(0+1) = 1

a1 = -(4)^(1) + 2^(1+1) = 0

a2 = -(4)^(2) + 2^(2+1) = -8

hence the explicit formua is : an = -(4)^n + 2^(n+1) , for n3 0

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