Question

3. a) If the solution of the recurrence relation αun−1 +βun−2 = f(n),(n ≥ 2) is

un = 1−2n+3.2

n

, then determine the values of α,β and f(n).

Answer 1

Given recurrence relation is-

$αu_{n−1} +βu_{n−2} = f(n)$

Also, $u_n=1-2n+3.2^n$

For solving such equation the value of coefficient is 1 so $\alpha=1,\beta=1$

The solution of the above equation can be written by characterstics root method as-

$u_n=c_1+c_2n+3.2^n$

The value of f(n) must be the value of the nin homogeneous part of the solution i.e. $f(n)=3.2^n$