I am lost with this HW from my DSP class! Z transform, Fourier Series

Hello to All,

I have fallen behind in my DSP class and am trying to catch up over our spring break. I am at my wit's end with this HW and would appreciate any help anyone can give me. I have attached the assignment .pdf as well as my work in trying to solve these three problems in another pdf... but I have not been able to solve any of the three problems

View attachment DSP HW4 draft.pdf -my work
View attachment DSP HW4.pdf -assignment

Thanks in advance,
Aaron

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If you are asking exactly where you are stuck, I believe you will get help from many people, but since you are posting the screenshots of the homework, I don't know how many people will reply you.. If you are lucky enough, you may get you problems solved from here. But people here will really appreciate you if you are asking specific doubts rather than posting the complete questions....
If I were you, I will definitely read the theory and will try to answer, and if not possible, I will ask doubts specifying exactly where I am stuck...

Ok, sorry I am new here.

y[n] + 2y[n-2] = x[n] - x[n-3] given difference equation

Starting from the first problem, I have used the given difference equation and set x[n]=delta[n] and y[n]=h[n] and calculated h[n] for n=1:10. From there I found the resulting pattern of h[n] to be...

h[n] = 2^(n/2)*cos(pi*n/2)*u[n] - 2^[(n-3)/2]*cos[pi*(n-3)/2]*u[n-3]

where u[n] is the unit step function

I now need to calculate H(z), so should I do this from h[n] or the given difference equation. When I tried the difference equation route, I found it difficult to perform partial fraction decomposition as the denominator did not factor out completely leading me to believe I should use "analysis"? to find H(z) from h[n]?

From what I understand I am supposed to
i. use "analysis" to find H(z)
ii. also use "analysis" to find FS coefficient of the periodic input signal x[n]=a_k
iii. find b_k=H(z=e^jkw_o)*a_k; b_k being FS coefficient of output
iiii. perform "synthesis" to find y[n] from b_k