jmc

Let's call the length of the shorter leg $x$. Then the length of the longer leg is $x+1$. Using the Pythagorean Theorem, we write the equation $x^2+(x+1{)}^2=29^2$ and solve for $x$. Expanding $(x+1{)}^2$, we get $x^2+x^2+2x+1=841$. This can be simplified to $2x^2+2x=840$, or $x^2+x=420$. Factoring out $x$ on the left, we can rewrite it as $x(x+1)=420$. In other words, the product of these two consecutive numbers is 420, which means they must each be close to the square root of 420. Indeed, $20\times 21=420$, so the legs must be 20 and 21. Their sum is $20+21=\boxed{41}$.