Geometric sequences and Fibbonacci Numbers

In summary, the conversation discusses the common ratio of a geometric sequence and finding the second and third terms of a geometric sequence. The common ratio of the Fibonacci sequence is (1+/-sqrt(5))/2 and the formula for finding the nth term is A((1+sqrt(5))/2)^n + B((1-sqrt(5))/2)^n. For the second problem, the second and third terms can be found by applying logic and using the formula for a geometric mean, which results in r=a^(1/3).
  • #1
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Homework Statement



A) In a certain geometric sequence every term is the sum of the two preceding terms, viz. the Fibonacci sequence, what can be said about the common ratio of the sequence?

So how do I go from 1,1,2,3,5,8,13,21,34... to (1+/-sqrt(5))/2?

Then find numbers A and B such (for all n) the nth term of the Fibonacci sequence is equal to:

A((1+sqrt(5))/2)^n + B((1-sqrt(5))/2)^n
B) The first term of a geometric sequence is 1 and the fourth term is a>0. Find the second and third terms.
2. The attempt at a solution

A) Unfortunately, I'm not even sure where to begin...

B) I know that the answer is a^(1/3) and a^(2/3). But figured it out just by thinking about the geometric mean being sqrt(ab), so I'm wondering how it would be done analytically
 
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  • #2
B)
You're given 1, ___, ___, a, ...

So you know that 1*r*r*r =a Then, r^3=a, r=a^(1/3) and your result follows.
 
  • #3
So its basically just applying some logic. Any ideas on the other problem, thanks.
 

What is a geometric sequence?

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant number. This constant number is called the common ratio, and it remains the same throughout the entire sequence.

What is the formula for finding the nth term of a geometric sequence?

The formula for finding the nth term of a geometric sequence is an = a1rn-1, where an is the nth term, a1 is the first term, and r is the common ratio.

How do you determine if a sequence is geometric?

A sequence is geometric if there is a common ratio between each term. This means that each term can be found by multiplying the previous term by a constant number.

What are Fibonacci numbers?

Fibonacci numbers are a sequence of numbers where each term is the sum of the two previous terms. The sequence starts with 0 and 1, and continues by adding the previous two numbers together (0+1=1, 1+1=2, 1+2=3, 2+3=5, and so on).

What are some real-life applications of geometric sequences and Fibonacci numbers?

Geometric sequences can be used to model population growth, compound interest, and depreciation of assets. Fibonacci numbers can be found in nature, such as in the arrangement of tree branches and the spiral of a snail's shell. They are also used in computer algorithms and coding.

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