1. Calculate $\sqrt{(31)(30)(29)(28)+1}$.
\begin{aligned} \text { 1. } & \because(k+1) \cdot k \cdot(k-1) \cdot(k-2)+1 \\ & =[(k+1)(k-2)][k(k-1)]+1 \\ & =\left(k^{2}-k-2\right)\left(k^{2}-k\right)+1 \\ & =\left(k^{2}-k\right)^{2}-2\left(k^{2} \cdots k\right)+1 \\ & =\left[\left(k^{2}-k\right)-1\right]^{2} \\ \therefore \quad & v^{\prime}(31) \cdot(30) \cdot(29) \cdot(28)+1 \\ & =30^{2}-30-1=869 .\end{aligned}