Sequences and Series problem

In summary, the sequence U1, U2, U3, ...Un... is related by Un = Un-1 +2Un-2 where n is greater or equal to 1, U1=2 and U2 =5. U7, U11, and U14 can be found by calculating U3 and U2 respectively.
  • #1
kalistella
4
0
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!:smile:
 
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  • #2
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.
 
  • #3
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:
Code:
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.
 
  • #4
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.:eek:

Cheers!
 
  • #5
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:
Code:
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.
 
Last edited:

1. What is the difference between a sequence and a series?

A sequence is a list of numbers or elements in a specific order, while a series is the sum of a sequence. In other words, a series is the result of adding all the terms in a sequence together.

2. How do I determine the pattern in a sequence or series?

To determine the pattern in a sequence or series, you can look for relationships between the terms, such as common differences or common ratios. You can also use algebraic equations to represent the pattern.

3. What are the different types of sequences and series?

There are several types of sequences and series, including arithmetic, geometric, and harmonic. In an arithmetic sequence, each term is obtained by adding a constant value to the previous term. In a geometric sequence, each term is obtained by multiplying a constant value to the previous term. In a harmonic sequence, each term is the reciprocal of a corresponding term in an arithmetic sequence.

4. How can I find the sum of a series?

To find the sum of a series, you can use various methods such as the formula for the sum of a finite arithmetic or geometric series, or the partial sum formula for an infinite series. You can also use techniques like telescoping or splitting the series into smaller parts.

5. How are sequences and series used in real life?

Sequences and series are used in many real-life situations, such as in finance, population growth, and scientific research. For example, compound interest is a type of geometric series, and the Fibonacci sequence is often seen in patterns observed in nature. In science, sequences and series are used to model natural phenomena and make predictions based on patterns.

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