Using RSolve for recurrence equation

You are missing the initial condition Q[0] == Q0.

Q[k] /. RSolve[{Q[k] == Q[k - 1] + α (Subscript[r, k] - Q[k - 1]), Q[0] == Q0}, 
    Q[k], k][[1]] // FullSimplify

This is equivalent to

$$(1-\alpha )^k Q0+\sum _{K[1]=1}^k \alpha\ (1-\alpha )^{k-K[1]}\ r_{K[1]}$$

Clear["Global`*"]

Format[Q[k_]] := Subscript[Q, k];
Format[Q0] = Subscript[Q, 0];

Include the initial condition in RSolve

sol = (RSolve[
      {Q[k] == Q[k - 1] + \[Alpha] (Subscript[r, k] - Q[k - 1]), Q[0] == Q0},
      Q[k], k][[1]] // Simplify) /. K[1] -> i

Translate the index of summation

sol2 = ((sol /. Sum -> Inactive[Sum]) /.
    Inactive[Sum][expr_, {i, imin_, imax_}] :>
     Inactive[Sum][(expr /. i -> i - 1), {i, imin + 1, imax + 1}]) //
  Collect[#, Q0] &

Simplify the summation term

(sol3 = sol2 /. (expr1_ * Inactive[Sum][expr2_, {i, imin_, imax_}]) :> 
     Inactive[Sum][expr1*expr2, {i, imin, imax}]) // Activate