fraction cancelling

a ppt is here

sometimes, as in question (7) above, only part-cancelling a fraction helps to find the nth term rule
sometimes, for other fraction sequences, there is a stray fraction that cancels - not part of the general cancelling rule
some features of an overall generalisation are relatively easy to discern
and others aren't (for me anyway)
in questions (8) and (9) above, for example,  you need to persevere in writing out a list...

what is neat about the last two sheets is that the general form for the uncancelled sequence
is equivalent to the general form for the cancelled sequence

e.g. for the sheet with 3 questions:
question (1) has a generalisation 5n / (20n - 5) for the uncancelled fractions
and n / (4n - 1) for the cancelled fractions

whilst not a surprise, it's a neat consideration of how to cancel down algebraic fractions