- #1
What is the formula for solving this Series/Sequence Problem?
- Thread starter ae4jm
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In summary, the conversation discusses a series/sequence problem and finding a formula for the sequence without having to calculate 1000 values. Various methods are suggested, including using arithmetic sequences and Euler's method. Eventually, a formula of a_n = 2n(n-1)+4 is derived and the conversation ends with a thank you for the help.
- #2
slider142
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Write out the formula for a1, a2, and a3 without summing and find the pattern so that you can write a formula for an that does not refer to an-1. It should remind you of a simple type of sum you already know a shortcut for.
- #3
HallsofIvy
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{an} is defined, recursively, by a1= 4, [itex]a_{n+1}= a_n+ 4n[/itex]. Find a1000.
The first thing I would do is start calculating a few values (hoping I won't have to go up to 1000!).
a2= 4+ 4(1), a3[/sup]= 4+ 4(1)+ 4(2), a4[/sup]= 4+ 4(1)+ 4(2)+ 4(3).
Hmmm, looks to me like an= 4(1+ 2+ 3+ ...(n-1)) so a1000= 4(1+ 2+ 3+ ... + 999). Can you find 1+ 2+ 3+ ...+ 999? It's an arithmetic sequence with common difference 1. Or you could use "Euler's method".
The first thing I would do is start calculating a few values (hoping I won't have to go up to 1000!).
a2= 4+ 4(1), a3[/sup]= 4+ 4(1)+ 4(2), a4[/sup]= 4+ 4(1)+ 4(2)+ 4(3).
Hmmm, looks to me like an= 4(1+ 2+ 3+ ...(n-1)) so a1000= 4(1+ 2+ 3+ ... + 999). Can you find 1+ 2+ 3+ ...+ 999? It's an arithmetic sequence with common difference 1. Or you could use "Euler's method".
- #4
tiny-tim
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Another method: rewrite the equation as:
Add them … what do you get … ? And then … ?
an+1 - an = 4n;
so an+2 - an+1 = 4(n+1).
so an+2 - an+1 = 4(n+1).
Add them … what do you get … ? And then … ?
- #5
ae4jm
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Gentlemen, I'm totally stuck. I sat here for the last 1.5 hrs and tried to figure this one out. I think that I'm making it too difficult. I've looked at the sequence all the way up through a6 and I'm just not seeing the formula.
I do know that to find the sum from 1 to 1000, would be (1000(1001))/2 or (n(n+1))/2. But I've tried finding the an like this also...dang I'm making this tuff aren't I?
I do know that to find the sum from 1 to 1000, would be (1000(1001))/2 or (n(n+1))/2. But I've tried finding the an like this also...dang I'm making this tuff aren't I?
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- #6
tiny-tim
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Hi ae4jm!
Humour me … what's an+2 - an? … what's an+3 - an?
Humour me … what's an+2 - an? … what's an+3 - an?
- #7
ae4jm
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I got 12 and 24?
- #8
tiny-tim
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Hi ae4jm!
Yes … that's a3 - a1 and a4 - a1.
But you'll get more of an idea of what's going on if you answer the general question:
what's an+2 - an? … what's an+3 - an? (in terms of n)
Yes … that's a3 - a1 and a4 - a1.
But you'll get more of an idea of what's going on if you answer the general question:
what's an+2 - an? … what's an+3 - an? (in terms of n)
- #9
ae4jm
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I believe that the first one is 6-4=2 and 7-4=3, right?
- #10
ae4jm
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does this look correct for
Code:
[tex]a_{n}=4(\frac{n(n-1)}{2})+4[/tex]- #11
tiny-tim
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Woohoo! And now, of course, simplify it to 2n(n - 1) + 4.
Any questions?
- #12
ae4jm
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Thanks! That was a dandy, for me anyways! I'm sure this practice will help me recognize these sequences a little better in the future. Thanks for sticking in there with me.
1. How do you solve a series/sequence problem?
To solve a series/sequence problem, you need to first identify the pattern or rule governing the sequence. This can be done by looking at the differences between consecutive terms, the ratio between consecutive terms, or any other noticeable pattern. Once the pattern is identified, you can use it to find the missing terms or to continue the sequence.
2. What is the difference between a series and a sequence?
A series is the sum of the terms in a sequence, while a sequence is a list of numbers that follow a specific pattern or rule. In other words, a series is the result of adding the terms in a sequence, while a sequence is a list of numbers that can be added together to form a series.
3. What is the importance of series and sequences in mathematics?
Series and sequences are important in mathematics because they are used to model real-world situations and to solve various mathematical problems. They also help in understanding and proving mathematical concepts and theories, as well as in calculating sums and limits.
4. How can I recognize the type of series/sequence problem?
There are different types of series and sequences, such as arithmetic, geometric, and Fibonacci. To recognize the type of problem, you need to look at the pattern or rule governing the sequence. For example, if the difference between consecutive terms is constant, it is an arithmetic sequence, while if the ratio between consecutive terms is constant, it is a geometric sequence.
5. Can series/sequences be infinite?
Yes, series and sequences can be infinite. An infinite sequence is a sequence that continues indefinitely, while an infinite series is a series whose sum can be calculated even though it has an infinite number of terms. Examples of infinite series include geometric series and Taylor series.
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