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(B) Given the recurrence relation $2a_{n+2} - 5a_{n+1} + 3a_n = 0$.The characteristic equation is $2x^2 - 5x + 3 = 0$,which factors as $(2x - 3)(x - 1

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$3a_{100} - 100$

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(B) Given the recurrence relation $2a_{n+2} - 5a_{n+1} + 3a_n = 0$.The characteristic equation is $2x^2 - 5x + 3 = 0$,which factors as $(2x - 3)(x - 1) = 0$.Thus,the roots are $x = 1$ and $x = \frac{3}{2}$.The general term is $a_n = A(1)^n + B(\frac{3}{2})^n = A + B(\frac{3}{2})^n$.Using initial conditions:For $n = 0: A + B = 0 \Rightarrow A = -B$.For $n = 1: A + \frac{3}{2}B = \frac{1}{2}$ $\Rightarrow -B + \frac{3}{2}B = \frac{1}{2}$ $\Rightarrow \frac{1}{2}B = \frac{1}{2}$ $\Rightarrow B = 1, A = -1$.So,$a_n = (\frac{3}{2})^n - 1$.Now,$\sum_{k=1}^{100} a_k = \sum_{k=1}^{100} ((\frac{3}{2})^k - 1) = \sum_{k=1}^{100} (\frac{3}{2})^k - \sum_{k=1}^{100} 1$.The sum of the geometric series is $\frac{\frac{3}{2}((\frac{3}{2})^{100} - 1)}{\frac{3}{2} - 1} = 3((\frac{3}{2})^{100} - 1)$.Therefore,$\sum_{k=1}^{100} a_k = 3((\frac{3}{2})^{100} - 1) - 100 = 3a_{100} - 100$.

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

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