Fibonacci Sequence Use the result of Problem 86 to do the following problems. List the first 11 terms of the Fibonacci sequence. List the first 10 terms of the ratio u n + 1 u n As it gets large. what number does the ratio approach? This number is referred to as the golden ratio. Rectangles whose sides are in this ratio were considered pleasing to the eye by the Greeks. For example, the facade of the Parthenon was constructed using the golden ratio. Write down the first 10 terms of the ratio u n u n + 1 As n gets large, what number does the ratio approach? This number is referred to as the conjugate golden ratio. This ratio is believed to have been used in the construction of the Great Pyramid in Egypt. The ratio equals the sum of the areas of the four face triangles divided by the total surface area of the Great Pyramid. Fibonacci Sequence Let u n = ( 1 + 5 ) n − ( 1 − 5 ) n 2 n 5 Define the n th term of a sequence. Show that u 1 = 1 and u 2 = 1 . Show that u n + 2 = u n + 1 + u n . Draw the conclusion that { u n } is a Fibonacci sequence.
Fibonacci Sequence Use the result of Problem 86 to do the following problems. List the first 11 terms of the Fibonacci sequence. List the first 10 terms of the ratio u n + 1 u n As it gets large. what number does the ratio approach? This number is referred to as the golden ratio. Rectangles whose sides are in this ratio were considered pleasing to the eye by the Greeks. For example, the facade of the Parthenon was constructed using the golden ratio. Write down the first 10 terms of the ratio u n u n + 1 As n gets large, what number does the ratio approach? This number is referred to as the conjugate golden ratio. This ratio is believed to have been used in the construction of the Great Pyramid in Egypt. The ratio equals the sum of the areas of the four face triangles divided by the total surface area of the Great Pyramid. Fibonacci Sequence Let u n = ( 1 + 5 ) n − ( 1 − 5 ) n 2 n 5 Define the n th term of a sequence. Show that u 1 = 1 and u 2 = 1 . Show that u n + 2 = u n + 1 + u n . Draw the conclusion that { u n } is a Fibonacci sequence.
Solution Summary: The author explains how to determine the first two terms of the Fibonacci sequence.
Fibonacci Sequence Use the result of Problem 86 to do the following problems.
List the first 11 terms of the Fibonacci sequence.
List the first 10 terms of the ratio
u
n
+
1
u
n
As it gets large. what number does the ratio approach? This number is referred to as the golden ratio. Rectangles whose sides are in this ratio were considered pleasing to the eye by the Greeks. For example, the facade of the Parthenon was constructed using the golden ratio.
Write down the first
10
terms of the ratio
u
n
u
n
+
1
As
n
gets large, what number does the ratio approach? This number is referred to as the conjugate golden ratio. This ratio is believed to have been used in the construction of the Great Pyramid in Egypt. The ratio equals the sum of the areas of the four face triangles divided by the total surface area of the Great Pyramid.
Fibonacci Sequence Let
u
n
=
(
1
+
5
)
n
−
(
1
−
5
)
n
2
n
5
Define the
n
th
term of a sequence.
Show that
u
1
=
1
and
u
2
=
1
.
Show that
u
n
+
2
=
u
n
+
1
+
u
n
.
Draw the conclusion that
{
u
n
}
is a Fibonacci sequence.
Answer all the following questions.
1.) Without adding, find the sum of the first 15 terms of the Fibonacci sequence. Explain and show your solution.Give atleast brief explanation each step.
2.)Without looking at the sequence, what is F15 if F14 is 377? Explain and show your solution. Give atleast brief explanation each step.
3.)The ratio Fib(n+1)/ Fib n, as n gets larger is said to approach the Golden Ratio, which is approximately equal to 1.618. What happens to the inverse of this ratio, Fib n/Fib(n+1)? What does this quantity approach? How does this compare to the original ratio?Give atleast brief explanation each step.
Answer parts a, and b of the following question.
8. a) The sum of the first 18 terms of an arithmetic series is 513, and the 18th term of the series is 52. What is the first term of the series?
b) The sum of an arithmetic series is 192. The value of the first term is 5, and the value of the nth term is 27. Find the number of terms.
Given the explicit formula for an arithmetic sequence find the first five terms and the term named in the problem. 7) an = -11 + 7n
Find a34
8) an = 65 - 100n
Find a39
Chapter 11 Solutions
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
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