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On a clean sheet of paper, compute the 3rd and 5th Fibonacci numbers using Binet's formula.

On A Clean Sheet Of Paper Compute The 3rd And 5th Fibonacci Numbers Using Binets Formula class=

Sagot :

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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

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