JEE Math Practice Question

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

1. A.$3 a_{100}+100$
2. B.$3 a_{100}-100$
3. C.$3 a_{99}-100$
4. D.$3 a_{99}+100$