We define a sequence ${a_n}$: $$a_1=1,a_{n+1}=\sqrt{a_n+n^2},n=1,2,...$$(1)Find $\lfloor a_{2019}\rfloor$ (2)Find $\lfloor a_{1}^2\rfloor+\lfloor a_{2}^2\rfloor+...+\lfloor a_{20}^2\rfloor$
Source: 2019 Taiwan APMO Preliminary
Tags: algebra, floor function, Sequence, recursive
We define a sequence ${a_n}$: $$a_1=1,a_{n+1}=\sqrt{a_n+n^2},n=1,2,...$$(1)Find $\lfloor a_{2019}\rfloor$ (2)Find $\lfloor a_{1}^2\rfloor+\lfloor a_{2}^2\rfloor+...+\lfloor a_{20}^2\rfloor$