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

I need to show that: ns^2_{n+1} = (n-1)s^2_n + \frac{n}{n+1}(x_{n+1} - \bar{x}_n)^2

where \bar{x} is the mean, and s^2_n is the variance for a set of data with n elements, and s^2_{n+1} is the variance for the same set of data, with 1 additional element ( x_{n+1}) 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.

some formulas
\bar{x} = \frac{\sum_{i=1}^n x_i}{n}

s^2 = \frac{\sum_{i=1}^n(x_i-\bar{x})^2}{n-1}

or, alternatively

s^2 = \frac{\sum_{i=1}^n x_i^2 - \frac{(\sum_{i=1}^n x_i)^2}{n}}{n-1}

In a different portion of question, they also mentioned that the mean of the set with x_{i+1} added to it can be derived from the current mean. I determined that this formula is \bar{x}_{i+1} = \frac{n\bar{x}_n + x_{n+1}}{n+1}