Solve the recurrence relation an = 5an-1 – 6an-2 with initial conditions a= 1 and a= 2.

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Given recursion relation

an = 5an-1 – 6an-2

Here, degree  = 2

Now, the characteristics Equation is

r2 = 5r – 6

or, r2 – 5r + 6 = 0

or, (r-2)(r-3) = 0

We get, r = 2,3

Let say r1 = 2 and r2 = 3

Noe, the general solution of the homogeneous equations

an = A1r1n + A2 r2n Where, A1 and A2 are scalar

or, an = A12n + A23n   ……….. (i)

Given initial contions

a0 = 1 🡲 n = 0

a0 = A120 + A230

or, A1 + A2 = 1  ——— (ii)

Also, a1 = 2 🡲 n = 1

a1 = A121 + A231

2A1 + 3A2 = 2  ———-(iii)

Solving Equation (ii) and (iii)

A2 = 0 and A1 = 1

From Equation, the general solution is

an = 2n + 0 × 3n

an = 2n

 

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