Binet's Formula
Given in 1943 by Jacques Philippe Marie Binet. An explicit formula used to find the nth term of the Fibonacci sequence. The formula is f_n = \frac{1}{\sqrt{5} }
5
1
[\frac{1 + \sqrt{5} }{2}
2
1+
5
ⁿ - [\frac{1 - \sqrt{5} }{2}
2
1−
5
ⁿ. n is the number of terms in the Fibonacci sequence. This is derived from the general form of quadratic equation.
Solutions:
a. Given: Binet's Formula: f_n = \frac{1}{\sqrt{5} }
5
1
[\frac{1 + \sqrt{5} }{2}
2
1+
5
ⁿ - [\frac{1 - \sqrt{5} }{2}
2
1−
5
ⁿ
n = 29
f_n = \frac{1}{\sqrt{5} }
5
1
[\frac{1 + \sqrt{5} }{2}
2
1+
5
ⁿ - [\frac{1 - \sqrt{5} }{2}
2
1−
5
ⁿ
Find the 29th term of the Fibonacci sequence.
f₂₉ = \frac{1}{\sqrt{5} }
5
1
[\frac{1 + \sqrt{5} }{2}
2
1+
5
²⁹ - [\frac{1 - \sqrt{5} }{2}
2
1−
5
²⁹
f₂₉ = \frac{1}{2.2360679775}
2.2360679775
1
[\frac{1 + 2.2360670775}{2}
2
1+2.2360670775
²⁹ - [\frac{1 - 2.2360679775}{2}
2
1−2.2360679775
f₂₉ = .447213595[\frac{3.2360679775}{2}
2
3.2360679775
²⁹ - [\frac{-1.2360679775}{2}
2
−1.2360679775
²⁹
f₂₉ = .447213595 [(1.61803399)²⁹ - (-.618033989)²⁹]
f₂₉ = (.447213595)(1,149,851.6190675)
f₂₉ = 514,229
Find the 30th term of the Fibonacci sequence.
f₃₀ = \frac{1}{\sqrt{5} }
5
1
[\frac{1 + \sqrt{5} }{2}
2
1+
5
³⁰ - [\frac{1 - \sqrt{5} }{2}
2
1−
5
³⁰
f₃₀ = \frac{1}{2.2360679775}
2.2360679775
1
[\frac{1 + 2.2360670775}{2}
2
1+2.2360670775
³⁰ - [\frac{1 - 2.2360679775}{2}
2
1−2.2360679775
³⁰
f₃₀ = .447213595[\frac{3.2360679775}{2}
2
3.2360679775
³⁰ - [\frac{-1.2360679775}{2}
2
−1.2360679775
³⁰
f₃₀ = .447213595 [(1.61803399)³⁰ - (-.618033989)³⁰]
f₃₀ = (.447213595)[1860498.04 - (-.0000000537490506)]
f₃₀ = (.447213595)(1860498.04)
f₃₀ = 832,040
b. Find the 31st term of the Fibonacci sequence.
31st term = 29th term + 30th term
f₃₁ = f₂₉ + f₃₀
f₃₁ = 514,229 + 832,040
f₃₁ = 1,346,269
What is the Binet's Formula: https://brainly.ph/question/4959269
#LearnWithBrainly
