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Solve a systems of equations with a sum in it?

Anna Ebert
Posted 6 years ago

COMPLETE EDIT:

I am trying to solve a symbolic equation and I've heard from some people that Mathematica is able to do it and from others that it isn't possible. It is actually quite easy to solve by hand but I want to learn Mathematica and I would be great if Mathematica is capable of doing something like this.

The problem is the following:

enter image description here

Mathematica Code

Subscript[X, t] == 
    (1 - \[Theta])*Sum[\[Theta]^(k - 1)*(Subscript[P, t - k] - Subscript[P, t - k - 1]), {k, 1, t - 1}] + 
     \[Theta]^(t - 1)*1; Subscript[NN, t] == Subscript[X, t]/(\[Gamma]*\[Sigma]^2); 
   Subscript[M, t] == (Subscript[D, t] - (T - t - 1)*\[Gamma]*\[Sigma]^2*Q - Subscript[P, t])/(\[Gamma]*\[Sigma]^2); 
   u*Subscript[NN, t] +  \[Mu]*Subscript[M, t]==Q; u == \[Mu] - 1; 
   Inequality[1, Greater, \[Mu], GreaterEqual, 0

People have been saying that it is possible, suggest that I should use RSolve but I do not fully understand the function. The guide on Mathematica says the following:

RSolve[eqn,a[n],n]

so for eqn I should put my equations in with a comma as separator. a[n] are my variables I think. n should be t and k.

Trying to do so doesn't give me the result I want though unfortunately.

All in all my problem does not only concern these equations I posted. I want to learn how to solve symbolic equations in general with Mathematica.

Thanks!
POSTED BY: Anna Ebert
2 Replies
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Anna Ebert
Anna Ebert
Posted 6 years ago

Hello,

thank you for your help.

If I understand correctly, you only used the first equations and solved this one for Pt. However what about the other equations?
POSTED BY: Anna Ebert
Bill Nelson
Bill Nelson
Posted 6 years ago

Mathematica sometimes can't say much about "abstract" sums, where the upper bound is unknown.

This

x[1]=1;
Table[{t,(1-?)*Sum[?^(t-1)(P[t-k]-P[t-k-1]),{k,1,t-1}]},{t,2,6}]//Simplify

shows this

{{2, (?-1)*?  *(P[0] - P[1])},
 {3, (?-1)*?^2*(P[0] - P[2])},
 {4, (?-1)*?^3*(P[0] - P[3])},
 {5, (?-1)*?^4*(P[0] - P[4])},
 {6, (?-1)*?^5*(P[0] - P[5])}}

which implies this

x[t]==(?-1)?^(t-1)(P[0]-P[t-1])

which says this

P[t-1]== -x[t]/((?-1)?^(t-1))+P[0]

If you have some additional information about P[0] then this might help you write the definition of P[t]

Please check every step of this carefully
POSTED BY: Bill Nelson
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