Can you find the general formula for A_n+3?

In summary, the conversation discusses finding the general formula and demonstration by induction for the equation A_n = A_{n-1} + 2n. The formula is given as A_{n+3} = 3A_{n+2} - 3A_{n+1} + A_n, with the three initial values of A_1 = 3, A_2 = 7, and A_3 = 13. Various steps and suggestions are discussed, including using induction to prove the formula and factoring out a common term in the summation. The conversation also includes a proof for the statement A_{n+1} = A_n + 2(n+1).
  • #1
kezman
37
0
Hi I have many problems trying to find the general formula and the demonstration by induction.
Let
[tex] A_1 = 3[/tex]
[tex] A_2 = 7[/tex]
[tex] A_3 = 13[/tex]
A_n+3 = 3A_n+2 - 3A_n+1 + A_n
I could only find this.
A_n+1 = A_n + 2n
 
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  • #2
What sort of class are you in?


A_n+1 = A_n + 2n
How did you arrive at this? It can't be right, though: it doesn't work for n=1 or n=2.

Oh, I bet you meant [itex]A_n = A_{n-1} + 2n[/itex], or [itex]A_{n+1} = A_n + 2(n+1)[/itex]. (Incidentally, you could try proving this formula by induction, to make sure you're on the right track)


One trick that's often useful is to not do arithmetic. If [itex]A_n = A_{n-1} + 2n[/itex], then write [itex]A_2 = 3 + 2\cdot2[/itex] instead of [itex]A_2 = 7[/itex].
 
  • #3
sorry I am in my way of learning Latex.
I mean:
[itex]A_n = A_{n-1} + 2n[/itex]
And the formula given in the problem is:
[itex]A_{n+3} = 3A_{n+2} - 3A_{n+1} + A_n[/itex]
With the three A1 A2 A3 defined.
Ive been trying but I couldn't find the general formula.
 
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  • #4
kezman said:
...I mean:
[itex]A_n = A_{n-1} + 2n[/itex]
So far so good.
Now, you can follow Hurkyl's suggestion:
A1 = 3
A2 = A1 + 2 . 2 = 3 + 2 . 2
A3 = A2 + 2 . 3 = 3 + 2 . 2 + 2 . 3
A4 = 3 + 2 . 2 + 2 . 3 + 2 . 4
A5 = 3 + 2 . 2 + 2 . 3 + 2 . 4 + 2 . 5
A6 = 3 + 2 . 2 + 2 . 3 + 2 . 4 + 2 . 5 + 2 . 6
...
An = 3 + 2 . 2 + 2 . 3 + 2 . 4 + 2 . 5 + 2 . 6 + 2 . 7 + ... + 2 . n
Now, do you see anything that can be factored out?
Can you go from here? :)
 
  • #5
For
[itex] A_n = 3 + \sum\limits_{i = 2}^n 2i [/itex]
then
[itex] A_1 = 3 [/itex]
Is this correct?
 
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  • #6
If you can prove it by induction, it's correct. :smile:
 
  • #7
Induction:
1)
[itex] A_1 = 3 + \sum\limits_{i = 2}^1 2i = 3 [/itex]
(Is this OK?)
2) I had this result :[itex]A_{n+1} = A_n + 2(n+1)[/itex]

With [itex] A_n = 3 + \sum\limits_{i = 2}^n 2i [/itex] =>
[itex]A_{n+1} = 3 + \sum\limits_{i = 2}^{n+1} 2i[/itex] (the upper index of the summatory in this case is (n+1))
=> [itex] A_{n+1} = 3 + \sum\limits_{i = 2}^n 2i + 2(n+1)[/itex]
=> [itex] A_{n+1} = A_n + 2(n+1) [/itex]

Is this correct?
 
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  • #8
Well, you've done it backwards!

[tex] A_{n+1} = A_n + 2(n+1) [/tex]

is the statement you know to be true, whereas

[tex]A_{n+1} = 3 + \sum\limits_{i = 2}^{n+1} 2i[/tex]

is the statement you were trying to prove.

(given the assumption that [itex] A_n = 3 + \sum_{i = 2}^n 2i [/itex])



(Actually, I'm assuming you've already given a proof of [itex] A_{n+1} = A_n + 2(n+1) [/itex] is correct given the original recurrence relation. If you have not yet done so, then you should work with the original recurrence)
 
  • #9
Yes is true.I still have to prove first [tex] A_{n+1} = A_n + 2(n+1) [/tex]
 

1. What is the general formula and why is it important to find it?

The general formula is a mathematical expression that represents a pattern or relationship between variables. It is important to find the general formula because it allows us to make predictions, solve problems, and gain a deeper understanding of a given phenomenon.

2. How do you go about finding the general formula?

The process of finding the general formula involves analyzing data, identifying patterns and relationships, and using mathematical techniques such as algebra, calculus, or statistics to derive a mathematical expression that represents the relationship between variables.

3. Can the general formula be used to solve any problem?

No, the general formula is specific to a particular phenomenon or set of data. It is not a universal solution to all problems, but rather a tool for understanding and solving problems related to a specific context or situation.

4. Are there any limitations to using the general formula?

One limitation of the general formula is that it may not account for all variables or factors that can affect a phenomenon. It also relies on the assumption that the relationship between variables is consistent and predictable, which may not always be the case in real-world situations.

5. How can the general formula be applied in real-world scenarios?

The general formula can be applied in various fields such as physics, chemistry, economics, and engineering to make predictions, solve problems, and develop new theories. It can also be used in data analysis and decision-making processes to gain insights and inform decision-making.

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