Sequence output by a Galois type LFSR I have provided a diagram of a Galois type LFSR. I am told it outputs a sequence of period 31.

Find the sequence output by this LFSR.

By applying the shifts I got the seemingly trivial answr of $0,0,0,0,1$ which I know must be wrong since it has a period of 31. Where have I gone wrong?
You made a mistake in the propagation of the feedback.

If you follow the LFSR very slowly this is what you get:

initial state:
+--------------------------------------------------------------+
|                         |           |           |            |
|    +-----+     +-----+  |  +-----+  |  +-----+  |  +-----+   |
|    |     |     |     |  v  |     |  v  |     |  v  |     |   |
+---&gt;+  0  +----&gt;+  0  +--+-&gt;+  0  +--+-&gt;+  0  +--+-&gt;+  1  +-------&gt; ...
     |     |     |     |     |     |     |     |     |     |
     +-----+     +-----+     +-----+     +-----+     +-----+

the 1 get on the branch and we are going to do the xor
+--------------------------------------------------------------+
|                         |           |           |            |
|    +-----+     +-----+  |  +-----+  |  +-----+  |  +-----+   |
|    |     |     |     |  1  |     |  1  |     |  1  |     |   |
+-1-&gt;+  ?  +--0-&gt;+  ?  +-0+-&gt;+  ?  +-0+-&gt;+  ?  +-0+-&gt;+  ?  +---1---&gt; ...
     |     |     |     |     |     |     |     |     |     |
     +-----+     +-----+     +-----+     +-----+     +-----+

compute the XORs
+--------------------------------------------------------------+
|                         |           |           |            |
|    +-----+     +-----+  |  +-----+  |  +-----+  |  +-----+   |
|    |     |     |     |  v  |     |  v  |     |  v  |     |   |
+-1-&gt;+  ?  +--0-&gt;+  ?  +--+1&gt;+  ?  +--+1&gt;+  ?  +--+1&gt;+  ?  +-----1-&gt; ...
     |     |     |     |     |     |     |     |     |     |
     +-----+     +-----+     +-----+     +-----+     +-----+


move the results to the next box (and output 1)
+--------------------------------------------------------------+
|                         |           |           |            |
|    +-----+     +-----+  |  +-----+  |  +-----+  |  +-----+   |
|    |     |     |     |  |  |     |  |  |     |  |  |     |   |
+---&gt;+  1  +----&gt;+  0  +--+-&gt;+  1  +--+-&gt;+  1  +--+-&gt;+  1  +-------&gt; ouput: 1
     |     |     |     |     |     |     |     |     |     |
     +-----+     +-----+     +-----+     +-----+     +-----+


So you have $10111$ instead of $10000$. Which will be then iterated and so on.