Question

A linear homogeneous recurrence relation of degree 2 with constant coefficients is a recurrence relation of the form an = Cian-1 + c2an-2, for real constants Ci and C2, and all n 2. Show that if an = r" for some constant r, then r must satisfy the characteristic equation, p2 - cir= c = 0.

Answer 1

rn for some constant r, r must be a root of the characteristic equation p2 - C1p - C2 = 0. That is, r must satisfy the characteristic equation.

To prove this, first assume an = rn for a constant r. Plugging this into a given recurring relation gives:

rn = C1rn-1 + C2rn-2

Expanding the right-hand side, we get:

rn = C1r(n-1) + C2r(n-2)

Simplifying by combining equal terms, we get:

rn - C1rn-1 - C2rn-2 = 0

This is a linear homogeneous recurrence relation of degree 2 with constant coefficients. In general, solutions to such recurrence relations are of the form an = ar1n + br2n. where r1 and r2 are the roots of the characteristic equation p2 - C1p - C2 = 0.

Therefore, if an = rn for some constant r, r must be a root of the characteristic equation p2 - C1p - C2 = 0. That is, r must satisfy the characteristic equation. This completes the proof.

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