Observe that from halfway between 1 and 2, which is , 
we go down from  by  to 1, then
we go up from  by  to 2, then
we go down from  by  to 1, then
we go up from  by  to 2, then
etc., etc., etc., ...

We know that the formula  goes down and up
-1,1,-1,1,-1,1,...

So we can use that as a multiplier of  to go down
and up from ,

So to get the 1st term we take  and add
, which is -1, times , to go down
to 1.

Then to get the 2nd term we take  and add
, which is +1, times , to go up
to 2.

Then to get the 3rd term we take  and add
, which is -1, times , to go down
to 1.

Then to get the 4th term we take  and add
, which is +1, times , to go up
to 2.

etc., etc., etc.,...

So when n is odd, we go down by  to 1 from 
and when n is even, we go up by  to 2 from . 

So to get the nth term we take  and add
 times ,

So the formula for the nth term, which we call  is



That will do for the formula, but maybe we want to write
it a little differently. We change  to an improper
fraction 



Then we can factor out 



Then we can change 'multiplying by ' to 
'dividing by 2' which is the same thing:



Edwin