In terms of n, 3, 7, 13, 27, 53, 107

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In summary, the explicit formula given is c1=1, c2=2, cn+1 = (cn + cn-1)/2 ; n>2. The top term in the sequence follows a pattern of 1, 2, 3, 7, 13, 27, 53, 107... and can be represented as a_{n+2}=2a_{n}+a_{n+1}. Additionally, the sequence can be rewritten in terms of a new variable a_n, where c_n = a_n/2^(n-2). By finding a recursion relation for a_n, a general solution can be determined.
  • #1
ziggie125
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Homework Statement



Find the explicit formula c1=1, c2=2, cn+1 = (cn + cn-1)/2 ; n>2


Homework Equations





The Attempt at a Solution



c1 = 1
c2 = 2
c3 = 3/2
c4 = 7/4
c5 = 13/8
c6 = 27/16
c7 = 53/32
c8 = 107/64

I know the bottom term is 2^n-2. I cannot find what the top is. If anyone sees it let me know thx.

Also if you know general form of 1, 1, -1, -1, 1, 1, -1, -1 ...

Thx a lot
 
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  • #2
The numerator kind of looks like [tex]a_{n+2}=2a_{n}+a_{n+1}[/tex] to me, so it would be like this:
c4 = (2*2) + 3 = 7
c5 = (2*3) + 7 = 13
c6 = (2*7) + 13 = 27
etc, it only works when n > 2 which seems to fit into what you mentioned as well. Hope that helps a bit!
 
  • #3
try multplying out explicitly for the first few terms - you should hopefully be able to pick up a pattern in terms of sums of powers of 2
[tex]3 = 1+2[/tex]
[tex]7 = 1 +2+2^2[/tex]
and follow on from there...
 
  • #4
I was told you cannot use the terms before it in the equation, ie. cn-1*2 + (-1)^n
 
  • #5
i'm not sure i know what you mean? try this though, write the c_n in term of a new variable a_n, for n>2
[tex]c_1 = 1[\tex]
[tex]c_2 = 2[\tex]
[tex]c_3 = \frac{3}{2} = \frac{a_3}{2}[\tex]
[tex]c_4 = \frac{7}{2^2} = \frac{a_4}{2^2}[\tex]

so you want to find the a_n, with
[tex]c_n = \frac{a_n}{2^{n-2}}[\tex]

how about seeing if you can find a recursion relation for the a_n, using the original... will be similar to what refraction posted, but the form should help lead to a general solution.
 

What is the pattern or rule for the sequence 3, 7, 13, 27, 53, 107?

The sequence follows the rule n^2 + n + 1, where n represents the position in the sequence.

What is the value of the 10th term in the sequence?

The 10th term in the sequence is 331, as it follows the pattern n^2 + n + 1 when n = 10.

Is there a limit to how far this sequence can continue?

As a scientist, I cannot definitively answer this question as it is based on an unknown pattern. However, based on the current pattern, the sequence can continue infinitely.

What is the significance of the numbers in this sequence?

The numbers in this sequence are all prime numbers, meaning they can only be divided by 1 and themselves. This could potentially have implications in number theory and cryptography.

How can this sequence be applied in real life or in other fields of study?

The sequence could potentially be applied in computer science, as prime numbers are often used in encryption algorithms. It could also have applications in number theory and mathematics in general.

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