I have three values
Mean (population mean, not from sample)
$\bar{x} = 3$
Number of elements in population (number of values in original set)
$n = 15$
A new value
$x_{(n+1)} = 7$
What is the correct formula to calculate the new mean? I would calculate it like this
$$ \bar{x}' = x_{(n+1)} * \frac{1}{(n+1)} + \bar{x} * \frac{n}{(n+1)} $$
which can be shortened
$$ = \frac{x_{(n+1)}}{(n+1)} + \frac{\bar{x} * n}{(n+1)} = \frac{x_{(n+1)} + \bar{x} * n}{(n+1)} $$
Is that the correct formula? Or did I miss something?
Yes, your formula works, and you can apply it to the case at hand.
$\bar x$ is the "starting" population mean, $n$ the population count, $\bar x'$ is the new population mean. Then
$$\bar{x}' = \frac{\bar x \cdot n + x_{n+1}}{n+1} = \frac{3\cdot 15 + 7}{15+1} = \frac{52}{16} = 3.25$$