Related Recurrence Relations

Algebra Level 5

Let ana_n, bnb_n, and cnc_n satisfy the system of recursive relations below for n2n \geq 2:

{bn2n2=cn1cn23an22n3+an15an2=2bn1+cn2cn6an2+5an1=an2n2 \large \begin{cases} b_n - 2^{n-2} = c_{n-1} - c_{n-2} - 3a_{n-2} \\ 2^{n-3} + a_{n-1} - 5a_{n-2} = 2b_{n-1} + c_{n-2} \\ c_n - 6a_{n-2} + 5a_{n-1} = a_n - 2^{n-2} \end{cases}

If a0=5a_0 = 5, a1=10a_1 = 10, b0=25948b_0 = -\frac{259}{48}, b1=658b_1 = -\frac{65}{8}, c0=74c_0 = \frac{7}{4}, and c1=32c_1 = \frac{3}{2}, find the value of a7+b5c6a_7 + b_5c_6.

This problem is part of the set "Xenophobia"

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