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Old 22nd May 2013, 08:21 PM   #1 (permalink)
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Recurrence relation given. Find formula for nth term. What is the best way?

A sequence u0, u1, u2, ... is such that u(n+1) = 0.9u(n) + 90.
Show that u(n) = (0.9^n)(u0 - 900) + 900.

This question is quite easy. I'm only wondering whether I have the most efficient technique. My solution is shown below. Would appreciate if someone could show me a more efficient way of doing this particular question, assuming there is one. (Not counting math induction.)

What I do is find the pattern:

u0 = u0
u1 = (0.9)u0 + 90
u2 = (0.9^2)u0 + (0.9)90 + 90
u3 = (0.9^3)u0 + (0.9^2)90 + (0.9)90 + 90
.
.
.
u(n) = (0.9^n)u0 + (0.9^n-1)90 + (0.9^n-2)90 + ... + (0.9)(90) + 90

After this it's easy, because the first term is (0.9^n)u0 and the rest are just n terms of a G.P. So it's easy to sum everything and the correct answer is obtained.

But is there a "cleaner" (more elegant) way of finding u(n) for this particular example? Appreciate any tips.

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Old 31st May 2013, 04:49 PM   #2 (permalink)
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Re: Recurrence relation given. Find formula for nth term. What is the best way?

Actually, i think your method is the most straightforward way for most people/students. Mine is not...

For me i considered:

Let:
So, i will get:


So i will get a GP in general with:

Now consider:

I will than use gemoteric series and method of difference to produce:

Upon simplification:

You should kinda realise by now that this method is very similar to the one proposed.


Last edited by megaman123; 31st May 2013 at 04:51 PM.
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Old 5th June 2013, 09:48 PM   #3 (permalink)
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Re: Recurrence relation given. Find formula for nth term. What is the best way?

Eh, thanks for this! Took me a while to see how you got that first line so quickly:

but once I saw it, it was quite obvious.

I've seen the MoD being applied in this way before, so I could follow the rest very quickly. It's just what I was looking for, more or less! Very elegant, this way. You one cool dude la. But I doubt I can teach this to my students cos it would be too much for the average student.

But anyway, from what I can see, you can use this method to find u(n) whenever the recurrence relation is of the linear form form u(n) = Au(n-1) + B. That's great, I'll use this method the next time.

Thanks again!

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