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Find the explicit form of u_n

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October 7, 2010
algebrapolynomialalgebra unsolved

Problem Statement

Let {un}\{u_n \} be the sequence defined by its first two terms u0,u1u_0, u_1 and the recursion formula un+2=un−un+1.u_{n+2 }= u_n - u_{n+1}. (a) Show that unu_n can be written in the form un=αan+βbnu_n = \alpha a^n + \beta b^n, where a,b,α,βa, b, \alpha, \beta are constants independent of nn that have to be determined.
(b) If Sn=u0+u1+⋯+unS_n = u_0 + u_1 + \cdots + u_n, prove that Sn+un−1S_n + u_{n-1} is a constant independent of n.n. Determine this constant.
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