Solution to Suppose a recurrence relation an=7an−1−12an−2 where a1=16 and a2=52 can be represented in explicit formula, … - Sikademy
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Archangel Macsika

Suppose a recurrence relation an=7an−1−12an−2 where a1=16 and a2=52 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. **If the explicit formula is in the form of Formula 2, consider p < q. Determine p and q

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 characteristic equation:

x^2-7x+12=0

x=\frac{7\pm\sqrt{49-48}}{2}

x_1=3,x_2=4


solution:

a_n=p\cdot 3^n+q\cdot 4^n

a_1=3p+4q=16

a_2=9p+16q=52


4q=4 \implies q=1

p=(16-4q)/3=4


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