Variable in summation w/ unknown # of iterations

[Guest]

Find [tex]T_{total}[/tex], such that [tex]T_{total}=\sum_{x=1}^{n}T_{n}[/tex], where
[tex]L_{tot}=\pi\sum_{x=1}^{n}T_{x}(A+(2x-1)B)[/tex]
[tex]T_{max}(x-1)<T_{x}\le T_{max}x[/tex]

where [tex]L_{tot}[/tex], [tex]A[/tex], [tex]B[/tex], & [tex]T_{max}[/tex] are known but [tex]n[/tex] is not.


I can make a table in excel to do this but requires some trial & error and then fine-tuning to get the best answer

9kOJqDM.png (28.78 KiB) Viewed 2031 times

I'd like to simplify in a way that [tex]T_{total}[/tex] can be calculated without a table or T&E

[Guest]

OP here,

I should note that [tex]T_{x}[/tex] must reach its maximum (depending on iteration) before starting to increase [tex]T_{x+1}[/tex]
such that [tex]T_1\ge T_2\ge ... \ge T_n[/tex]

Imagine filling multiple small yet varying sized beakers with a large volume of water, you can only start filling the next beaker when the last one is full. [tex]T_{tot}[/tex] would be the sum of the height of the liquid level in each beaker

[Guest]

gave up on excel, found FreeMAT as a tool, similar to MATLAB without the price tag
I gave the script an arbitrary n=50 but it could be 10E5 for all I care, as the while loop will exit once the calculation reaches Ltot
(T will be to the nearest integer that allows the calc to reach Ltot as a min)

Code: Select all

for x=1:50; C(x)=(A+(2*x-1)*B); end
Lcalc=0;Lint=0;i=1;T=zeros(1,50);
while (Lcalc<Ltot)
    T(i)=T(i)+1;
    Lcalc=Lint+T(i)*C(i);
    if (T(i)==Tmax)
        Lint=Lcalc;
        i=i+1;
    end
end
Ttot=sum(T)