Troubleshooting the Partition Function for a System with 3 Spins

annaphys
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Homework Statement
Consider 3 spins s1,s2,s3 on a solid lattice. The spins can have the values -1 or +1. Find the partition function.
Relevant Equations
H = J1*s1*s2 + J2*s2*s3. J1,J2 >0
I'm having problems solving the partition function. I've attached a photo of where I currently am. Am I on the right track? What should be my next step?
 

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I'm not an expert in this field but just on symmetry considerations, I would expect there to be a "j_3s_1s_3" term in it.
 
The hamiltonian is given.
 
Solved it. Thought about it too hard. Just need to work out the sums and simplify. One gets then two cosh() functions.
 

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