nth terms of a continuing, regular pattern

if you counted pencils from the start (as 1)
and when you finish a row you keep counting, in the next row
and so on (for ever)

in what number positions will there be a dark blue pencil?

how many would you need to count beyond a million to a position where the pencil would be dark blue?

same thing, with coloured eggs
this is a 'window' of a regular pattern that continues downwards for ever and ever...
what positions do the blue eggs occupy?
the yellow ones?

same thing, with stripes
which continue to the right (or left) for ever and ever...

what positions do the dark blue stripes occupy?






links with modular arithmetic