If α,β are the roots of the equation x2−x+q=0 and Sr=αr+βr, find Sn in terms of ∑aiSn−i, where ai are constant terms for each Sn−1.

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Draidayerabauu · Accepted Answer

Step 1Note that the roots satisfy&amp;nbsp;x2=x−q. In particular, they satisfyxn=xn−1−qxn−2for&amp;nbsp;n≥2. In matrix form, we may write this asαnαn-1=1-q10αn-1αn-2;&amp;nbsp;&amp;nbsp;n≥2.Step 2Inductively, we getαnαn-1=1-q10n-1α1;&amp;nbsp;&amp;nbsp;n≥2.And similarly, we have the same for&amp;nbsp;β. Adding the similar equation for&amp;nbsp;β&amp;nbsp;to the above equation gives usSnSn-1=1-q10n-112;&amp;nbsp;&amp;nbsp;n≥2.Step 3Thus, we have[Sn]=101-q10n-112;&amp;nbsp;&amp;nbsp;n≥2.This is a &quot;closed form&quot; for a lenient enough definition. If you are not satisfied, you can do better by diagonalising (if&amp;nbsp;α≠β) the above matrix and getting a better form for it.(However, that will not be particularly helpful since it would involve&amp;nbsp;αn&amp;nbsp;and&amp;nbsp;βn.)

If \alpha, \beta are the roots of the equation x^2-x+q=0

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