If α and β are the roots of the equation x2−ax+b=0 and An=αn+βn, then which of the following is true?

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An+1=aAn+bAn−1

An+1=bAn+aAn−1

An+1=aAn−bAn−1

An+1=bAn−aAn−1

answer is C.

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Detailed Solution

An+1=αn+1+βn+1aAn−bAn−1=aαn+βn−bαn−1+βn−1 Now α+β=a,αβ=b∴ aAn−bAn−1=(α+β)αn+βn−αβαn−1+βn−1=αn+1+αβn+αnβ+βn+1−αnβ−αβn=αn+1+βn+1=An+1∴ An+1=aAn−bAn−1