filters - A general conceptual understanding of moving average

I have a conceptual question here. I understood how to calculate a Moving Average(MA).

For example :

x(n) = 0, 1, 2, 3, 4, 5

MA = 0, 1/2, 3/2, 5/2,7/2, 9/2, where for calculation of the MA we only take the past values without any update

Is there also a case , where we use the newly calculated values and use that to update the MA for the next time instant?

x(n) = 0, 1, 2, 3, 4, 5

MA (n=0) = 0

MA (n=1) = (0 +1) /2 = 1/2

MA(n = 2) = (2 + 1/2) /2 = 5/4

MA(n = 3) = (3 + 5/4) = 17/4 and so on ?

Does this find an application anywhere?

Please help me with this understanding.

Answer

That system you're referring to is a first-order recursive filter

$$y[n]=\alpha x[n]+(1-\alpha)y[n-1],\qquad 0<\alpha<1\tag{1}$$

where $n$ is the (integer) time index, $y[n]$ is the output, and $x[n]$ is the input. You chose $\alpha=\frac12$.

That filter is also called exponentially weighted moving average. It has theoretically infinite memory and weighs the past input values with exponentially decreasing weights.