Rewrite the determinant of each triangular number in terms of combination (choose any eg 3,6,10,15) and, hence, establish a determinant for the nth triangular number with elements in terms of n...

Rewrite the determinant of each triangular number in terms of combination (choose any eg 3,6,10,15) and, hence, establish a determinant for the nth triangular number with elements in terms of n only. Show, by proof that this determinant always produces a triangular number.

Expert Answers
embizze eNotes educator| Certified Educator

The elements of Pascal's triangle can be written as:

`_0C_0`

`_1C_0` ` ``_1C_1`

`_2C_0` `_2C_1` `_2C_2`

`_3C_0` `_3C_1` `_3C_2` `_3C_3`

etc...

The triangular numbers are found along the diagonal beginning at `_2C_0` through `_3C_1` ,`_4C_2` ,`_5C_3` ,etc...

A rule for the triangular numbers using combinations is `_nC_(n-2)`

An algebraic rule for the triangular numbers is `(n(n-1))/2` : this generates all triangular numbers for `n>=1,n in NN` .

We show that `_nC_(n-2)=(n(n-1))/2` :

`_nC_(n-2)=(n!)/((n-2)!(n-(n-2))!)`

`=(n(n-1)(n-2)!)/((n-2)!2!)`

`=(n(n-1))/2` as required.