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I'm trying to figure out a way to approach sequences problems. So far, I can get them if I try enough different random formulas, but was hoping there's a better way to approach these problems.

For example, take this sequence:


I can see that the above sequence has a recurrence form of:


How would I find the close form of this sequence? What approach can I use that I can then apply to all other sequences?


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Arbitrary sequences can't be expected to have a neat closed form for their general element. Some exceptions are arithmetic sequences and geometric sequences for which general closed formulae are well known. Another class of sequences that admit a general technique for finding (thought it can be hard) closed forms is the class of sequences defined by linear recursive rules. A famous example is the Fibonacci sequence: $a_1=a_1=2$ and $a_{n+2}=a_{n+1}+a_n$. More general linearly recursive sequences look like $a_1=c_1$, $a_2=c_2$, and $a_{n+2}=\alpha\cdot a_{n+1}+\beta \cdot a_n$ for arbitrary $c_1,c_2,\alpha,\beta \in \mathbb C$. The most general linear recursive sequence is when one uses the $k$ previous terms to define the next one (the case above is $k=2$).

Any linearly recursive sequence can be described in terms of a matrix that is constructed from the coefficients in the defining relation. Thus, for a suitable matrix one has $\vec{v_{k+1}}=M\cdot \vec{v_k}$ and the elements in the vectors $\vec{v_k}$ generate the elements $a_k$ of the sequence. Then often closed formulae for the $a_k$ can be found by diagonalizing the matrix $A$. One such successful attempt is to get a closed form for the Fibonacci sequence. This method can also be used to derive properties of the given sequence (by looking, e.g., at the trace and determinant of the matrix).

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