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

u_{n} = 7^{n}-(-5)^{n}

satisfies the Cassini-type identity

u_{n-1}u_{n+1}-(u_{n})^{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 7^{n}-(-5)^{n} into u_{n-1}, u_{n+1}, and (u_{n})^{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…

Rob,

Good site, i too have been studying MS221 started to write the TMA i am stuck on the Cassini’s identity, like you follow up to the point “then after cancelling”. I have been using mathcad and the law of indices but still missing the trick, hence started searching the internet and found this site, i followed the link nad understand the example but the exercise in our books differs slightly could you possible give some pointers to the example in exercise book A

Any help would be appreciated

Michael

Hi Michael,

I’ve written up my notes on how I eventually managed to do Exercise 5.1 (b) from the Exercise Book A, which are here: https://catbear.files.wordpress.com/2009/03/ex-book-a-ex5-1b.doc . Hope that helps!

Rob

Thanks Rob.

hi rob

Many thanks for setting up this site, i am studying this years MS221course and came across this site while trying to find information on Cassini identities. Your notes have been very helpful as i was going around and around trying to understand this.

Wayne

Thanks Wayne, it’s great to hear that the notes have been helpful! I get quite a lot of visits from people searching for Cassini identity stuff, so it looks like loads of people (me included) found the course text a bit unclear on that topic.

top marks for your site, struggled with this and your notes are brilliant. Thanks so much for leaving them up for future OU students!!

I am so glad I am not alone here! I feel as though whoever wrote the course material for this missed out about five steps, I have no idea how they get from what seems a logical row of happy little numbers to the answers they have provided. I feel as though I have been going round in circles for days. I will definitely be having a look at your link Rob. And if all else fails I am going to go on the basis that what is good for the book is good for me and just write ‘then after cancelling’ and put the given answer. you never know.