Question

find the recurrence relation for Yn=A3^n+B(-4) ^n​

Answer 1

The recurrence relation is

$\sf y_{n + 2} + y_{n + 1} - 12y_{n} = 0$

Given :

The solution

$\sf y_{n} = A \: {3}^{n} + B \: {( - 4)}^{n}$

To find :

The recurrence relation

Solution :

Step 1 of 3 :

Write down the given Solution

Here the given Solution is

$\sf y_{n} = A \: {3}^{n} + B \: {( - 4)}^{n}$

Step 2 of 3 :

Find the characteristic equation

From above we see that 3 and - 4 are the roots of the characteristic equation

So the characteristic equation is

$\sf {\lambda}^{2} - ( - 4 + 3)\lambda + ( - 4 \times 3) = 0$

$\sf \implies {\lambda}^{2} + \lambda - 12 = 0$

Step 3 of 3 :

Find recurrence relation

Hence the required recurrence relation is

$\sf y_{n + 2} + y_{n + 1} - 12y_{n} = 0$