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Why am i getting different answers and which one is correct??enter image description here

In this also enter image description here

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Your first line is wrong. The partial fraction expansion of $X(z)$ should be :

$$ X(z) = \frac{ z (2z-\frac{5}{6})}{ (z-\frac{1}{2})(z-\frac{1}{3})} = 2 + \frac{ \frac{1}{2} }{z - \frac{1}{2}} + \frac{ \frac{1}{3}}{z - \frac{1}{3}} $$

Then they should yield the same inverse (causal) transform as:

$$ x[n] = 2\delta[n] + (1/2)^{n} u[n-1] + (1/3)^{n} u[n-1] $$

which is identical to : $$ x[n] = (1/2)^n u[n] + (1/3)^n u[n] $$

as obtained by the second method.

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  • $\begingroup$ Ok got it but what about second question? $\endgroup$
    – Aman Sinha
    Feb 29 '20 at 21:01
  • $\begingroup$ your second question is allright. You should see that $$3^n u[n] - 2^n u[n] = 3^n u[n-1] - 2^n u[n-1] $$ for all $n$. $\endgroup$
    – Fat32
    Mar 1 '20 at 9:15

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