# Substitution of sequences to derive recursive mean

I would have the following derivation to calculate the recursive moving average and I'd like to follow it as closely as possible in Mathematica.

For the sequence where $$y(i)$$ is considered as an observation of a at the ith instant:

$$y(i) = a + e(i); i = 1,2,...,N$$

The mean of the entire sequence is $$\widetilde{a}$$:

$$\widetilde{a}(k) = \frac{1}{k}\sum\limits_{i=1}^{k}{y(i)}$$

At for the $$(k - 1)^{th}$$ instant:

$$\widetilde{a}(k-1)=\frac{1}{k-1}\sum\limits_{i=1}^{k-1}{y(i)}$$

I am trying to use Mathematica to rearrange the above equations to obtain the following equation for the recursive mean:

$$\widetilde{a}(k) = \widetilde{a}(k-1)+\frac{1}{k}[y(k)-\widetilde{a}(k-1)]$$

I have the definitions of for each of the expressions:

OverTilde[a][k_, y_] := (1/k) Sum[y[i], { i, 1, k}]
eq1 = OverTilde[a][k, y]
eq2 = OverTilde[a][k - 1, y]


How do I substitute eq1 into eq2 in order to obtain the equation for the recursive mean?

OverTilde[a][k_, y_] := (1/k) Sum[y[i], {i, 1, k}];

which returns True. The reason for rule is that Mathematica currently can not do this automatically.