How Do You Solve a Recursive Sequence Problem with Given Initial Conditions?

[kalistella]

Hi

I have a problem with sequences and series. Can anybody help, please?

The question is

For the sequence U1, U2, U3, ...Un... the terms are related by
Un = Un-1 +2Un-2
where n is greater or equal to 1, U1=2 and U2 =5.

Find the values of U7, U11, and U14.

Can someone explain to me how to do it?

Thanks!

 

[acm]

This method is extremely crude:
U3= U2 + 2U1
U3= 9
Keep on going until you get 6 and 5.
For U11 and U14, Keep on going until you get 9, 10 and 13, 12 respectively.

 

[0rthodontist]

It isn't hard to calculate these by hand or write a program to calculate it. In fact, probably the simplest way to do it is to write a haskell program:

u 1 = 2
u 2 = 5
u n = u (n-1) + 2 * u (n-2)

save that in a file, then load it into the ghci interpreter and type in u 14.

Anyway, the question is do you just need to find the values by any means, or do you actually want to solve the recursion? If you only want to find the values then a simple way is to just write down u1 and u2, and from those compute u3. Then from u2 and u3 compute u4, and so on--not too hard if you have a calculator at hand. If you want to solve the recursion you need other methods.

 

[kalistella]

Sequences continued

Hi

Thanks for you help.

How about when U0=4 U1=-1

Un - Un-1 - 2Un-2=0

I'm a little thrown by the U=0.

Cheers!

 

[0rthodontist]

In that case you would have, for example, U2 = U(2-1) + 2 * U(2-2) = U(1) + 2 * U(0) = -1 + 2 * 4 = 7

By the way, the code I mentioned earlier is inefficient if you want to calculate say u 90. If you memoize it:

tab = [u n | n <- [0..]]

u 1 = 2
u 2 = 5
u n = tab ! (n-1) + 2 * tab ! (n-2)

it reads almost as good (tab is a table (a list), [u n | n <- [0..]] could be read as "the list of all u n such that n is a nonnegative integer" and ! is how you index a list). Then you can type in u 9000 and you'll get your answer in a couple seconds. I love haskell... it's too bad I don't have a good use for it yet besides little things like this.