Let $$G(x) = \frac{\exp(-ax) + \exp(-bx)}{2}$$
I need to find the coefficients $a$ and $b$.
Given: A table of 10 values of x and their corresponding G(x) values.
I need some help in figuring out how to go about the sum.
PS: The actual problem is pertaining to Dynamic Light Scattering.
As coffeemath commented, this is a typical nonlinear regression; it does not make any problem except that, in general, you need to provide reasonable estimates for parameters $a$ and $b$.
If you do not have any idea about these, what you could do is to build a regular grid over the parameters and compute for each pair the value of $$SSQ(a,b)=\sum_{i=1}^{10} \Big(\frac{e^{-ax_i} + e^{-bx_i}}{2}-G(x_i)\Big)^2$$ and look for the minimum value of this function; the best point will correspond to the starting values to use. All of this preliminary step can be done using Excel.
Now, use a nonlinear regression tool. If you do not have any, continue with a more and more refined grid procedure.