A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j\ge1$, once the bee reaches point $P_j$, she turns $60^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. How many inches is $P_{24}$ from $P_0$?

Call the $j$-th flight (from $P_{j-1}$ to $P_j$) step $j$; it has length $j$ and direction $60^\circ(j-1)$ counterclockwise from east.

1. Steps $j$ and $j+3$ point in opposite directions, and step $j+3$ is $3$ inches longer. Together they displace the bee by $3$ inches in the direction opposite to step $j$.
2. So six consecutive steps $j,\dots,j+5$ give a displacement of $3$ inches opposite to each of the directions of steps $j,j+1,j+2$ added together. For $j=6m+1$ these directions are $0^\circ,60^\circ,120^\circ$, whose unit vectors add to $(1,\sqrt3)$. So each block of six steps gives the displacement $-3(1,\sqrt3)$, of length $6$.
3. The $24$ steps form $4$ such blocks, all with the same displacement, so $P_{24}$ is at distance $4\cdot6=\boxed{24}$ inches from $P_0$.