These are essentially the same formula. If you add

to both sides of the first formula, you get

so

, which is the second formula with n replaced by by n-1 (i.e. all three subscripts decreased by 1).

But let's take your two formulas and do some algebra:

adding

to the first gives:

and substituting into the second gives:

and subtracting

from both sides gives:

.

That's the second formula you were looking for.

Now, if you take:

and instead of substituting for

, you substitute for

using

, you get:

, which is the first formula you were looking for.

- Hollywood