Spent several hours trying to prove this, and now my eyes are swimming :/

I need to show that:

where is the mean, and is the variance for a set of data with n elements, and is the variance for the same set of data, with 1 additional element ( ) added to it. The goal being, I assume, to calculate the new variance based on the old variance without having to redo the grunt work.

edit:

some formulas

or, alternatively

In a different portion of question, they also mentioned that the mean of the set with added to it can be derived from the current mean. I determined that this formula is