What is the general formula for the terms in this sequence?

  • Thread starter aruwin
  • Start date
  • Tags
    Notation
In summary, The sequence in question is a,b,x1,x2,...xn,..., where each term is equal to the arithmetic mean of its two preceding numbers. By substituting a and b, it can be observed that the denominator follows a power of 2 pattern, while the numerator follows a Jacobsthal sequence. The general formula for xn is x_n = \gamma_n(\alpha_n a + \beta_n b), where \gamma_n = 2^{-n}, \alpha_n = \beta_{n-1}, and the coefficients follow the Jacobsthal sequence.
  • #1
aruwin
208
0
I have a problem on how to generalize this sequence.
Problem:
In sequence a,b,x1,x2,...xn,...,each term is equals to the arithmetic mean of its two preceding numbers.Using a and b, find:

1.xn

2.lim(n->∞) xn

My working:

x1 = (a+b)/2, x2=(b+x1)/2
x3 = (x2+x1)/2...

Ok, so I did the same thing as above until x5 and substituted all that are needed to be substituted with a and b.And here's what I get:

(a+b)/2,(a+3b)/4,(3a+5b)/8,(5a+11b)/16,(11a+21b)/32+...

I realize a pattern here. Obviously, the denominator is just power of 2. I see a pattern in the numerator too,but it's hard to generalize it using n.I realize that the value of the second a is equals to the value of the previous b and the value of the third a is equals to the second b and it goes on.
You see it too,don't you?When the first term has b in the numerator,then the second term has a, and when the 2nd term has 3b in the numerator, the 3rd term has 3a and so on...now we just have to generalize that,don't we?
 
Last edited:
Physics news on Phys.org
  • #2
[itex]x_n = \gamma_n(\alpha_n a + \beta_n b)[/itex]

you have figured out that [itex]\gamma_n = 2^{-n}[/itex]

You've noticed that [itex]\alpha_n = \beta_{n-1}[/itex]

The pattern of the coefficients is 0, 1, 1, 3, 5, 11, 21, ... looks kinda familiar doesn't it?
Almost a Fibonacci series but actually it's the Jacobsthal sequence.

[itex]J_n = J_{n-1} + 2 J_{n-2} \; : \; J_0 = 0, J_1 = 1[/itex]
 
Last edited:

What is progression notation?

Progression notation is a method of representing a sequence of numbers or values in a shortened and concise format. It is often used in mathematics and computer science to describe patterns and sequences.

How is progression notation written?

Progression notation is typically written using the symbol "n" as a variable to represent the position of a number or value in the sequence. The first number in the sequence is represented as n=1, the second number as n=2, and so on.

What are the different types of progression notation?

The three main types of progression notation are arithmetic, geometric, and harmonic. Arithmetic progression involves adding a constant value to each term in the sequence. Geometric progression involves multiplying a constant value to each term in the sequence. Harmonic progression involves dividing a constant value by each term in the sequence.

What are some common examples of progression notation?

Some common examples of progression notation include the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, ...), which is a type of arithmetic progression, and the powers of 2 (2, 4, 8, 16, 32, 64, ...), which is a type of geometric progression.

How is progression notation useful?

Progression notation is useful for quickly describing and understanding patterns and sequences. It also allows for efficient calculations and predictions based on the given sequence. Progression notation is also commonly used in computer algorithms and data structures.

Similar threads

Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
2K
  • Calculus
Replies
24
Views
3K
  • Calculus and Beyond Homework Help
Replies
1
Views
3K
  • Calculus and Beyond Homework Help
Replies
6
Views
3K
  • Calculus and Beyond Homework Help
Replies
9
Views
5K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
13
Views
6K
  • Calculus and Beyond Homework Help
Replies
24
Views
2K
Back
Top