Solution to Suppose a recurrence relation an=2an−1−an−2 where a1=7 and a2=10 can be represented in explicit formula, … - Sikademy
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Archangel Macsika

Suppose a recurrence relation an=2an−1−an−2 where a1=7 and a2=10 can be represented in explicit formula, either as: Formula 1: an=pxn+qnxn or Formula 2: an=pxn+qyn where x and y are roots of the characteristic equation. Determine p and q Answer: p : q :

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Let us solve the characteristic equation of the recurrence relation a_n=2a_{n−1}−a_{n−2}, which is equivalent to a_n-2a_{n−1}+a_{n−2}=0. It follows that the characteristic equation x^2-2x+1=0 is equivalent to (x-1)^2=0, and hence has the roots x_1=x_2=1. It follows that the solution of the recurrence equation is a_n=p\cdot 1^n+q\cdot n1^n=p+q\cdot n. Since a_1=7 and a_2=10, we conclude that 7=a_1=p+q and 10=a_2=p+2q. Therefore, q=3 and p=4.

Answerp=4,\ q=3.

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