Let’s concentrate what happens when we move from one jar to the next. Symbolically,

\[p(w,k+1) = p(w,k) \cdot \frac{m+1}{m+n+1} + (1 - p(w,k)) \cdot \frac{m}{m+n+1}\]

where \(p(w,k)\) is the probability of drawing a white ball from the jar numbered \(k\).

We know that,

\[p(w,1) = \frac{m}{m+n}\]

Then,

\[\begin{align*} p(w,2) & = \frac{m}{m+n} \cdot \frac{m+1}{m+n+1} + (1 - \frac{m}{m+n})\cdot \frac{m}{m+n+1}\\ & = \frac{m}{m+n} \cdot \frac{m+1}{m+n+1} + \frac{n}{m+n} \cdot \frac{m}{m+n+1}\\ & = \frac{m\cdot (m+n+1)}{(m+n)(m+n+1)}\\ & = \frac{m}{m+n} \end{align*}\]

Seeing that \(p(w,1) = p(w,2)\), and that there is no reason that the case will be different for any \(k>2\), we conclude that the probability of drawing a white ball from any jar is the same as the probability of drawing a white ball from the first jar, i.e., \(p(w,k) = m/(m+n)\) for all \(k\geq 1\).