Generalisation of Pattern (1 Viewer)

[Sy123]

Is it possible to find the number of white boxes (or black), in the nth triangle, if the triangles keep following the same pattern?
The question originally asked to find the number of white boxes in the 5th triangle, but that was given to year 6. Im just wondering if its possible to generalise this.


Ive gotten as close to:



This is the number of white boxes in the nth triangle if n is even, however if n is odd then it has the same number of white boxes as (n-1) (which is even and you can plug it into this formula)

However I dont know how to generalise the above statement, if its even possible

 

[Sindivyn]

I think you would have to incorporate (-1)^n somewhere; how, I'm not sure.

 

[Carrotsticks]

http://en.wikipedia.org/wiki/Triangular_number

 

[Sindivyn]

n/2(n+1) - 1/2((2n-1) - (-1)^n(2n-1))
I think that works from testing

 

[Sy123]

This time I think Ive got it (deleted last post)



Works for 4 and 5 too.

I now have to do proof of induction to verify it

 

[Sindivyn]

Yeah, that should work. Have fun proving by induction