Let $ $ be a sequence such that ${\mathrm{a}}_{0}=0,{\mathrm{a}}_{1}=\frac{1}{2}$ and $2{a}_{n+2}=5{a}_{n+1}-3{a}_{n},n=0,1,2,3,\ldots \ldots$ Then $\sum _{\mathrm{k}=1}^{100}{\mathrm{a}}_{\mathrm{k}}$ is equal to: [JEE Main 2025, 28 Jan (Shift 1)]

Question

Let $ <a_n>$ be a sequence such that ${\mathrm{a}}_{0}=0,{\mathrm{a}}_{1}=\frac{1}{2}$ and $2{a}_{n+2}=5{a}_{n+1}-3{a}_{n},n=0,1,2,3,\ldots \ldots$ Then $\sum _{\mathrm{k}=1}^{100}{\mathrm{a}}_{\mathrm{k}}$ is equal to: [JEE Main 2025, 28 Jan (Shift 1)]

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Let \( <a_n>\) be a sequence such that \({\mathrm{a}}_{0}=0,{\mathrm{a}}_{1}=\frac{1}{2}\) and \(2{a}_{n+2}=5{a}_{…

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Let \( &lt;a_n&gt;\) be a sequence such that \({\mathrm{a}}_{0}=0,{\mathrm{a}}_{1}=\frac{1}{2}\) and \(2{a}_{n+2}=5{a}_{…

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Let \( <a_n>\) be a sequence such that \({\mathrm{a}}_{0}=0,{\mathrm{a}}_{1}=\frac{1}{2}\) and \(2{a}_{n+2}=5{a}_{…