A positive integer \(n\) is a balancing number if \(1 +\dots + (n - 1) = (n + 1) + \dots + (n + r)\) holds with some positive integer \(r\). The problem of finding balancing numbers goes back to the work of \textit{R. Finkelstein} [Am.\ Math.\ Mon.\ 72, 1082--1088 (1965; Zbl 0151.03305)]. Further, recently several results have appeared on certain generalizations of the problem (see the references of the paper). In the paper the authors extend the problem to arithmetic progressions. Let \(a,b\) be coprime integers with \(a>0\) and \(b\geq 0\). If for some positive integers \(n\) and \(r\) we have \[ (a+b) +\dots + (a+(n - 1)b) = (a+(n + 1)b) + \dots + (a+(n + r)b) \] then \(a+nb\) is an \((a,b)\)-balancing number. Write \(B_m^{(a,b)}\) for the sequence of these numbers. The authors prove several effective and explicit finiteness results concerning the equation \(B_m^{(a,b)}=f(x)\) in integers \(m,x\) where \(f\) is a polynomial with rational coefficients, taking integer values at integers. The main tools of the proofs are the modular method, elliptic curves and Baker's method.