I would iteratively have to solve the following equation at iteration $n$:

$C = \sum_{1 \leq i \leq n}{e^{\frac{x_i}{T}}x_i}$ for $T$.

Each iteration $i$ an unknown $x_i$ will be observed and $C$ is a constant.

For the first iteration obtaining $T$ is trivial, but I have no clue how to do so for later iterations as it would involve logarithms of sums.

I could settle for a (crude) approximation, but I would like to limit the computation in each iteration, thus I would like to reuse $T$ or some other partial result (recursive formula) to avoid summing over all observations (as $n$ can become very large).

Thank you in advance for your help.