A number of linked rings, each $1$ cm thick, hang on a peg. The top ring has outside diameter $30$ cm, each subsequent ring's outside diameter is $1$ cm less, and the bottom ring has outside diameter $5$ cm. What is the distance from the top of the top ring to the bottom of the bottom ring?

1. As in the problem, the distance equals the sum of all the inside diameters plus the two thicknesses of the bottom ring. The outside diameters are $30,29,\dots,5$ ($26$ rings), so the inside diameters are $28,27,\dots,3$, with sum $\frac{(28+3)\cdot26}2=403$. In general, with top outer diameter $T$ and bottom outer diameter $B$, the distance is $\dfrac{(T+B-4)(T-B+1)}2+2$.
2. With $T=30,B=5$: $\dfrac{(31)(26)}2+2=403+2=\boxed{405}$.