Recursive Sequence "3, 5, -2, 7, -9, 16, -25, 41" How would I start in solving this recursive sequence?  Thanks!
Sequence: 3, 5, -2, 7, -9, 16, -25, 41, ...
 A: I would try to denote the problem using a matrix-formulation, with a guessed linear recursion involving for instance three elements:
$$ \begin{array} {rrrrrr}
& & & & x_1 & \\
& & & & x_2 & \\
& & & & x_3 & \\
--&--&--&-&-- \\
3&5&-2& &7\\
5&-2&7&=&-9 \\
-2&7&-9& &16\\

\end{array} $$
where the left matrix is P, the vector X contains the unknowns $x_1,x_2,x_3$ and the right vector is Q, with the obvious insertions. We ask: is the following matrix-equation solvable:
$$ P * X = Q $$ $$ X = P^{-1} * Q $$
thus first we check, whether P is invertible. Maybe we need only a 2x2-matrix P or possibly we need higher dimension, if the the result is not correct for the following members of the given sequence. 
In our example we come along with a 2x2 matrix P 
$$ \begin{array}{} P=\begin{pmatrix} 3&5 \\\ 5&-2 \end{pmatrix} & & 
P^{-1}= \begin{pmatrix}  2/31 & 5/31 \\\ 5/31 & -3/31  \end{pmatrix} \end{array}$$
and get $X = (1,-1)$ so that $$a(k+2)=x_1 a(k)+x_2 a(k+1) = 1 a(k) - 1 a(k+1)$$ 
A: How does:


*

*$-2$ relate to $3$ and $5$?

*$7$ relate to $5$ and $-2$?

*$-9$ relate to $-2$ and $7$?


etc. If we write $f_i$ to be the $i$-th term in your sequence, these questions are just asking you: How can you express $f_n$ in terms of $f_{n-1}$ and $f_{n-2}$? 
Note that you may always define the first few numbers in a recursion by hand.
(Of course this is not some general strategy, but if you have such a problem, the desired formula won't be something too complicated - here I just guessed it would involve the two previous terms somehow, but again in general, that's probably the first thing you should try.)
