ABC26GN0222 · Number System

$$\begin{aligned} & 5^{2}+6^{2}+\ldots \ldots \ldots+10^{2}+20^{2}=\left(1^{2}+2^{2}+3^{2}+\ldots \ldots \ldots+10^{2}\right)-\left(1^{2}\right. \\ & \left.+2^{2}+3^{2}+4^{2}\right)+400 \\ & =\frac{1}{6} n(n+1)(2 n+1)-(1+4+9+16)+400, \text { where } n=10 \\ & =\left(\frac{1}{6} \times 10 \times 11 \times 21\right)-30+400=(385-30+400)=755 . \end{aligned}$$