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Linear Recurrence Relations

Take your recursion skills to the next level. If you've got a recurrence relation but no computer, how can you find a closed form? What about asymptotic behavior? How fast do rabbits reproduce?

Level 5

         

Let \( k_1, k_2, \ldots, \) be a sequence that is recursively defined as \(k_{n+2} = k_{n+1} + 2 k_{n}\), for all \(n\geq 1 \), with \(k_1 = k_2 = 1\). The infinite sum, \(S = \frac{k_1}{7 ^1} + \frac{k_2}{7 ^2} + \frac{k_3}{7 ^3} \ldots\), is a fraction of the form \(\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. What is the value of \(a+b\)?

The sequence \(a_0,a_1,a_2,\dots\) satisfies \[a_{m+n} + a_{m-n} = \frac{1}{2}(a_{2m}+a_{2n})\]

for all non-negative integers with \(m\geq n\). If \(a_1 = 1\), determine the value of \[a_{1995}\]

Define a subset \(S\) of the first \(30\) positive integers to be uneven if, for all \(i \in S\), \(i+2 \notin S\). For example, \(\{1, 2\}\) is an uneven subset, while \(\{1, 2, 3\}\) is not. If \(N\) represents the number of uneven subsets, find the remainder when \(N\) is divided by \(1000\).

Notes:

The empty set is considered to be an uneven subset.

You may want use a calculator at the end.

\[ \large { \begin{eqnarray} a_{n} & =& 2a_{n-1} + 3 \\ b_{n} &=& b_{n-1} + b_{n-2} \\ \end{eqnarray} } \]

For natural number \(n\), consider the two recurrence relation above subjected to conditions, with \(a_1 = 0, b_{1} = b_{2} = 1 \).

If the value of expression \((a_{b_{2015}} + 3)(a_{b_{2016}} + 3)\) can be expressed as \(p^{b_{q}} \dfrac{r}{s}\), where \(b_{q}\) is one of the term in the recurrence relations sequence above and \(p, r\) and \(r, s\) are pairwise coprime integers.

Find the value of \((p+q+r+s) \bmod {673}\).

\[ \large a_{n+2} = (n + 3)a_{n+1} - (n + 2)a_{n} \]

For whole numbers \(n\), consider the recurrence relation defined as above with \(a_{1} = 1, a_{2} = 3\).

Find \(\displaystyle\bigg( \sum_{k=1}^{2015} a_{k} \bigg)\pmod{100}.\)

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