Solve the recurrence relations together with the initial conditions given. a. an = 7an−1 − 10an−2 , for n ≥ 2 , a0 = 2 , a1 = 1 .

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To solve the given recurrence relation an=7an−1−10an−2 for n≥2, with initial conditions a0=2 and a1=1, we can use the method of characteristic equations.Step 1: Find the characteristic equation by assuming a solution of the form an=rn.Substituting this into the recurrence relation, we get:rn=7rn−1−10rn−2.Dividing both sides by rn−2 (assuming r≠0), we have:r2=7r−10.Step 2: Solve the characteristic equation to find the roots.r2−7r+10=0.This quadratic equation can be factored as (r−5)(r−2)=0.So, the roots are r1=5 and r2=2.Step 3: Write the general solution of the recurrence relation.Since we have distinct roots, the general solution is given by:an=c1·r1n+c2·r2n,where c1 and c2 are constants determined by the initial conditions.Step 4: Apply the initial conditions to find the specific solution.Using the initial conditions a0=2 and a1=1, we can set up a system of equations:a0=c1·r10+c2·r20=c1+c2=2,a1=c1·r11+c2·r21=5c1+2c2=1.Solving this system of equations, we find c1=−1 and c2=3.Step 5: Write the specific solution.Substituting the values of c1 and c2 into the general solution, we obtain:an=−r1n+3r2n.In this case, r1=5 and r2=2, so the specific solution is:an=−5n+3·2n.Therefore, the solution to the given recurrence relation with the initial conditions a0=2 and a1=1 is an=−5n+3·2n.

Solve the recurrence relations together with the initial

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