This is a good question with more than one answer.  

You may find the following article helpful:

Y. Eidelman, I. Gohberg, V. Olshevsky, Eigenstructure of order-one-quasiseparable matrices, Linear Alg. and its Applications 405, 2005, 1-40:

http://ac.els-cdn.com/S0024379505001655/1-s2.0-S0024379505001655-main.pdf?_tid=eaa92722-8066-11e4-acfa-00000aacb35f&acdnat=1418214204_7c62bd0714372f3543ce3e7a478ad741

See Section 1.4, p. 6 for an overview.    An introduction to tridiagonal matrices starts in Section 1.1,   p. 2 (see the examples, starting on page  3).d

This question is probably too old now for any answer to be useful to the original poster.

But anyway, the best methods are correction and inspection: First one must get the problem right; in this case it seems obvious to me that the matrix entry in the upper right corner should be b instead of 1, and the entry in the lower left corner should be c instead of 1.

With that in place one notes, by inspection, that the problem is translation invariant. Hence an orthonormal set of eigenvectors are (discretised) plane waves,

Ψn(km) = exp(ikmn)/√N, with km = 2πm/N and m=0, 1, ..., N-1.

Here it is natural (but not mandatory) to let the vector index n run from zero, n = 0, 1, ..., N-1. The corresponding eigenvalues are

λ(k) = a + b e-ik + c eik.

In the unlikely case that the original problem is correct, considerable progress can still be made, analytically reducing the problem to a non-standard 2x2 matrix eigenvalue problem. I.e., with the eigenvalue parameter entering in a highly nonlinear manner, so for explicit results one must ultimately resort to numerical computations.

On the first line, the last element should be b or 0.? (a 1 complicates matters). Should be square matrix of course.

If the matrix is cyclic or circulant then all eigenvalues and eigenvalues may be found analitically.(loop translation)

If not the strictly tridiagonal case is one of the easiest cases to treat numerically. Any standard dioganizational

package will do.(linear translation)

Looks non symmetric, you get complex eigenvalues. b=c for symmetric matrix for real eigenvalues.