Actually, it’s not Cassini-type identities themselves that have been perplexing me, it’s the simplifying and cancelling involved in exercises like the one below:

Show, by substitution, that the sequence
un = 7n-(-5)n
satisfies the Cassini-type identity
un-1un+1-(un)2 = -144(-35)n-1
for n = 1,2,3,….

I always seem to get stuck at the same part of the process; I’m fine with the bit where we substitute the sequence into the left-hand side of the identity – in this example, substituting 7n-(-5)n into un-1, un+1, and (un)2 – but I have a lot of trouble manipulating the resulting expressions into the form required by the question.

I’ve spent a lot of time this week working through the solutions to these problems in Exercise Book A, and in each case I could follow the working right up until the line “Therefore, after cancelling”, which seems to conceal some crucial bit of manipulation that I just can’t see. To be honest, that line is starting to remind me of that “then a miracle occurs…” cartoon, except that the kind of cancellation in these exercises probably isn’t mysterious or miraculous to most MS221 students.

So I spent a few hours today trying to find similar exercises online, in the hope that the worked solutions would shed some light on the problem, but I didn’t have a great deal of luck. In fact, I was just about to give up and email my tutor to ask for help, when I found this brilliant MS221 Revision Aid. Thankfully the example question about Cassini-type identities had a worked solution that went into much more detail than the ones in Exercise Book A, and I was able to follow the working with very little trouble. I actually said “Hooray!” out loud, at that point!

I’m happy enough with my understanding of the exercises now that I can move on to the next chapter of the course, but I think I’ll probably make the effort to go to this week’s MS221 tutorial, and see if I can get my tutor to go through one of the examples from Chapter A1 in detail, just to make sure there’s nothing I’m missing. That’s if the dreaded snow doesn’t return to wreak havoc with the local bus service…