Calculate n-order determinant (n>=2):
$\left|\begin{array}{ccccc}x_{1}-a_{1} & x_{2} & x_{3} & \cdots & x_{n} \\ x_{1} & x_{2}-a_{2} & x_{3} & \cdots & x_{n} \\ x_{1} & x_{2} & x_{3}-a_{3} & \cdots & x_{n} \\ \vdots & \vdots & \vdots & & \vdots \\ x_{1} & x_{2} & x_{3} & \cdots & x_{n}-a_{n}\end{array}\right|$
$a_{i} \neq 0, i=1,2, \cdots, n$
The result should be
$(-1)^{n-1} a_{1} a_{2} a_{3} \cdots a_{n}\left(\sum_{i=1}^{n} \frac{x_{i}}{a_{i}}-1\right)$
I want to use the FindSequenceFunction
to get the result. Here is my code:
Clear["Global`*"];
Format[a[n_]] := Subscript[a, n];
Format[x[n_]] := Subscript[x, n];
NewMatrix[n_Integer?Positive] :=
Module[{i = 1, j = 1, M = Array[m, {n, n}]},
For[i = 1, i <= n, i++,
For[j = 1, j <= n, j++,
If[j == i, m[i, j] = x[j] - a[j], m[i, j] = x[j]]]]; M]
tab = Table[Det[NewMatrix[i]], {i, 2, 10}]
$\left\{a_{1} a_{2}-a_{2} x_{1}-a_{1} x_{2},-a_{1} a_{2} a_{3}+a_{2} a_{3} x_{1}+a_{1} a_{3} x_{2}+a_{1} a_{2} x_{3}, a_{1} a_{2} a_{3} a_{4}-a_{2} a_{3} a_{4} x_{1}-a_{1} a_{3} a_{4} x_{2}-a_{1} a_{2} a_{4} x_{3}-a_{1} a_{2} a_{3} x_{4},\right.$, $-a_{1} a_{2} a_{3} a_{4} a_{5}+a_{2} a_{3} a_{4} a_{5} x_{1}+a_{1} a_{3} a_{4} a_{5} x_{2}+a_{1} a_{2} a_{4} a_{5} x_{3}+a_{1} a_{2} a_{3} a_{5} x_{4}+a_{1} a_{2} a_{3} a_{4} x_{5},$, $a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}-a_{2} a_{3} a_{4} a_{5} a_{6} x_{1}-a_{1} a_{3} a_{4} a_{5} a_{6} x_{2}-a_{1} a_{2} a_{4} a_{5} a_{6} x_{3}-a_{1} a_{2} a_{3} a_{5} a_{6} x_{4}-a_{1} a_{2} a_{3} a_{4} a_{6} x_{5}-a_{1} a_{2} a_{3} a_{4} a_{5} x_{6},-a_{1} a_{2} a_{3} a_{4} a_{5} a_{6} a_{7}+$ $\left.\quad a_{2} a_{3} a_{4} a_{5} a_{6} a_{7} x_{1}+a_{1} a_{3} a_{4} a_{5} a_{6} a_{7} x_{2}+a_{1} a_{2} a_{4} a_{5} a_{6} a_{7} x_{3}+a_{1} a_{2} a_{3} a_{5} a_{6} a_{7} x_{4}+a_{1} a_{2} a_{3} a_{4} a_{6} a_{7} x_{5}+a_{1} a_{2} a_{3} a_{4} a_{5} a_{7} x_{6}+a_{1} a_{2} a_{3} a_{4} a_{5} a_{6} x_{7}\right\}$
Then,
FindSequenceFunction[tab, n]
FindSequenceFunction $[\left\{a_{1} a_{2}-a_{2} x_{1}-a_{1} x_{2},-a_{1} a_{2} a_{3}+a_{2} a_{3} x_{1}+a_{1} a_{3} x_{2}+a_{1} a_{2} x_{3}, a_{1} a_{2} a_{3} a_{4}-a_{2} a_{3} a_{4} x_{1}-a_{1} a_{3} a_{4} x_{2}-a_{1} a_{2} a_{4} x_{3}-a_{1} a_{2} a_{3} x_{4},\right.$, $-a_{1} a_{2} a_{3} a_{4} a_{5}+a_{2} a_{3} a_{4} a_{5} x_{1}+a_{1} a_{3} a_{4} a_{5} x_{2}+a_{1} a_{2} a_{4} a_{5} x_{3}+a_{1} a_{2} a_{3} a_{5} x_{4}+a_{1} a_{2} a_{3} a_{4} x_{5},$, $a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}-a_{2} a_{3} a_{4} a_{5} a_{6} x_{1}-a_{1} a_{3} a_{4} a_{5} a_{6} x_{2}-a_{1} a_{2} a_{4} a_{5} a_{6} x_{3}-a_{1} a_{2} a_{3} a_{5} a_{6} x_{4}-a_{1} a_{2} a_{3} a_{4} a_{6} x_{5}-a_{1} a_{2} a_{3} a_{4} a_{5} x_{6},-a_{1} a_{2} a_{3} a_{4} a_{5} a_{6} a_{7}+$ $\left.\quad a_{2} a_{3} a_{4} a_{5} a_{6} a_{7} x_{1}+a_{1} a_{3} a_{4} a_{5} a_{6} a_{7} x_{2}+a_{1} a_{2} a_{4} a_{5} a_{6} a_{7} x_{3}+a_{1} a_{2} a_{3} a_{5} a_{6} a_{7} x_{4}+a_{1} a_{2} a_{3} a_{4} a_{6} a_{7} x_{5}+a_{1} a_{2} a_{3} a_{4} a_{5} a_{7} x_{6}+a_{1} a_{2} a_{3} a_{4} a_{5} a_{6} x_{7}\right\},n]$
It doesn't work.
Sum
/Inactive[Sum]
. For instanceD[]
, maybeDSolve[]
, but I don't recallFindSequenceFunction
doing it. Is there an example? It might help with finding an approach to the problem. $\endgroup$