Note the wff \[ P(1) \land P(n)\rightarrow P(n+1) \] is not what the author of the following solution intended: it should be \[ P(1)\wedge({P(n)}\rightarrow{P(n+1)}) \]
I didn't get a satisfactory solution to the second part, so I just wrote in *most* of the details to a student's paper....
The second part is a little sloppy with the mix of L and S (and I note that the inequality should be replaced by ).
Part three appears to have been solved by inspection (recognition of the triangular numbers from problem 1, perhaps).