Solve Recursion Equation: (n-1)*a[n+1] - n*a[n] + 10*n = 0

[galois427]

how do you solve the recursion equation (n-1)*a[n+1] - n*a[n] + 10*n = 0?

 

[TenaliRaman]

You seem to have missed the initial conditions.
Apply any good looking transform ... say something like one sided z-transform

-- AI

 

[M98Ranger]

To solve this recursion equation, we need to first understand what it is asking for. A recursion equation is an equation that defines a sequence of values based on previous values in the sequence. In this case, the equation defines the values of a sequence a[n], where n represents the position in the sequence.

To solve this equation, we will use a technique called "unfolding." This involves expanding out the equation by substituting in the previous values of a[n] until we reach a known value. Let's start by expanding out the first few terms:

a[0] = (0-1)*a[1] - 0*a[0] + 10*0
a[1] = (1-1)*a[2] - 1*a[1] + 10*1
a[2] = (2-1)*a[3] - 2*a[2] + 10*2

We can continue this process until we reach a known value, such as a[0] or a[1]. For simplicity, let's start by solving for a[0]:

a[0] = (0-1)*a[1] - 0*a[0] + 10*0
a[0] = -a[1] + 0
a[0] = -a[1]

Now, let's substitute this value into the next equation:

a[1] = (1-1)*a[2] - 1*a[1] + 10*1
a[1] = 0*a[2] - a[1] + 10
a[1] = -a[1] + 10
2a[1] = 10
a[1] = 5

We can continue this process to solve for the rest of the values in the sequence. By substituting in the known values, we can eventually solve for a[n] in terms of n. In this case, we can see that a[n] = 10/n. Therefore, the solution to the recursion equation is a[n] = 10/n.

In summary, to solve a recursion equation, we use the technique of unfolding to expand out the equation and substitute in known values until we reach a solution for a[n] in terms of n.