Hi, use a sequence and not a loop and you can you use the add command and with Students[Statistics] the Mean command. 

AddMean.mw

There are a few ways to do what you want to do, but one thing I would suggest is change your function definition from q1(t) to q[1](t) or q_1(t) similarly with q2(t) or just have different functions in general, ie f(t),g(t),q(t) etc. How to do what you want can be found by two methods:

The easiest way in my oponion is to use the Physics package and the LagraneEquation command. If you want to "brute" for it you can as well, both situations are shown in the attached file. 

Lagrangian.mw

You can define your own coordinate system in a couple ways after loading physics pacakge

1) 

Coordinates(X=[t,r,theta,phi])

or 

Setup(coordinatesystems=[X=(t,r,theta,phi)])

and then when you call a metric it will organize into the form of your coordinates. Or you can define your own metric. 

Coordinates.mw

First of, Psi is inside of a ln, so when doing the command degree it fails. You would need to convert the expression into a polynomial in Psi first. 

Secondly you are solving for c but there is no c only c_1.  As well the system of equations you are trying to solve is overdetermined, You have 7 equations and 4 unkowns inlcuding more logarithmic functions so it even more complicated. Hope this helps. 

Degree.mw

Your question title mentioned GRTensorIII but then you spoke of the physics package in your post so I am unsure what you are interested in exactly but wtih the Physics package you can get specific geodesics and integrate numerically. I have attached a radial free fall solution to show you how this can be done. 

You can modify this to include more geodesics or different physical situations. 

If there is something unlcear feel free to ask. 

GeodesicsPhysics.mw

Set your digits at the top of your notebook and then define the matrix as a variable and evalf the variable? 

Unless you want something else with a little more sophistication. 

DigitsMatrix.mw

The numpoint command should come after you specify your ranges. See attached.

DEplot.mw