How to use the property of function to find the cumulative value of the function value?

Questions
1

The functions f[x] and g[x] have domains of R,and f[x]+g[4-x]=4,g[4]=8,g[x]-f[x-8]=8,and g[4+x]=g[4-x].Find the value of sum = Sum[f[2*m], {m, 1, 16}]

I tried this set of code and did not find the cumulative value

Clear["`*"]

g[x_ /; x <= 4] = g[x] = g[8 - x]
g[4] = 8

f[x_ /; x >= 4] = f[x] = 4 - g[4 - x]
f[x_ /; x < 4] = f[x] = g[x + 8] - 8
\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(m = 1\), \(16\)]\(f[2  m]\)\)
2

If you’re going to post the same question as you did on another forum, you really need to provide a link to that question as you can be wasting the time of folks trying to help you that don’t know there might be informative comments or even answers.

https://mathematica.stackexchange.com/questions/302375/how-to-find-the-cumulative-value-through-the-abstract-function-analytic-expressi

3

Here is a way:

With[{m = 16},
 vls = Solve[Flatten@Table[{
      f[x] + g[4 - x] == 4,
      g[4] == 8,
      g[x] - f[x - 8] == 8,
      g[4 + x] == g[4 - x]},
     {x, -2 m, 2 m}]];
 Sum[f[2 k], {k, m}] /. vls[[1]]]
4

Thank you! What is the meaning and function of this code?

Flatten@Table[{f[x] + g[4 - x] == 4, g[4] == 8, g[x] - f[x - 8] == 8, 
   g[4 + x] == g[4 - x]}, {x, -32, 32}]
5

It gives numerical values to the variable x and sets up the system of equations. Try the effect with a smaller range:

Flatten@Table[{f[x] + g[4 - x] == 4, g[4] == 8, g[x] - f[x - 8] == 8, 
   g[4 + x] == g[4 - x]}, {x, -1, 1}]