Let $\left\langle {{a_{\rm{n}}}} \right\rangle $ be a sequence such that ${a_0} = 0,{a_1} = \frac{1}{2}$ and $2{a_{{\rm{n}} + 2}} = 5{a_{{\rm{n}} + 1}} - 3{a_{\rm{n}}},{\rm{n}} = 0,1,2,3, \ldots $ Then $\sum\limits_{{\rm{k}} = 1}^{100} {{a_k}} $ is equal to
