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Fibonacci Sequence and Binet's Formula

Fibonacci Sequence

Have you ever counted a number of petals in a flower? You might think that any number is possible. But you might be surprised because nature seems to favor a particular numbers like 1, 2, 3, 5, 8, 13, 21 and 34. It may seem coincidence to you but it's actually forming a pattern - Fibonacci Sequence.

fibonacci-sequence-and-binets-formula

Fibonacci Sequence is popularized in Europe by Leonardo of Pisa, famously known as "Leonardo Fibonacci". Leonardo Fibonacci was one of the most influential mathematician of the middle ages because Hindu Arabic Numeral System which we still used today was popularized in the Western world through his book Liber Abaci or book of calculations. In this book, Fibonacci post and solve a problem involving the growth of population of rabbits based on idealized assumptions.

Fibonacci Sequence is a wonderful series of numbers that could start with 0 or 1. The nth term of a Fibonacci sequence is found by adding up the two Fibonacci numbers before it. For example, in the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13,...

2 is found by adding the two numbers before it, 1+1=2. Following the same pattern, 3 is found by adding 1 and 2, 5 is found by adding 2 and 3 and so on. But what if you are asked to find the 100th term of a Fibonacci sequence, are you going to add the Fibonacci numbers consecutively until you get the 100th term? Yes, it is possible but there is an easy way to do it. You can use the Binet's formula in in finding the nth term of a Fibonacci sequence without the other terms.

Binet's Formula

Binet's Formula is an explicit formula used to find the nth term of the Fibonacci sequence. It is so named because it was derived by mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre.

fibonacci-sequence-and-binets-formula

We can also use the derived formula below. This is the general form for the nth Fibonacci number.

fibonacci-sequence-and-binets-formula

Example 1: Find the 10th term of the Fibonacci sequence: 1, 1, 2, 3, 5, 8, ...

Answer: Since you're looking for the 10th term, n = 10.

fibonacci-sequence-and-binets-formula

Example 2: Find the 25th term of the Fibonacci sequence: 1, 1, 2, 3, 5, 8, ...

Answer: Since you're looking for the 25th term, n = 25.

fibonacci-sequence-and-binets-formula

Who is Jacques Philippe Marie Binet?

fibonacci-sequence-and-binets-formula

Jacques Philippe Marie Binet was a French mathematician, physicist, and astronomer born in Rennes. He died in Paris, France in 1856. He made significant contributions to number theory and the mathematical foundations of matrix algebra. In his memoir in the theory of conjugate axis and the moment of inertia of bodies, he enumerated the principle which is known now as Binet's Theorem. He is also recognized as the first to describe the rule for multiplying matrices in 1812 and most specially the Binet's Formula expressing Fibonacci numbers in close form is named in his honour, although the same result was known to Abraham de Moivre a century earlier.

Try this!

For each question, choose the best answer. The answer key is below.

  1. What is the 12th term of the Fibonacci sequence?
    • 89
    • 144
    • 233
    • 377
  2. Find the 32nd term of the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13,...
    • 832 040
    • 987 625
    • 1 346 269
    • 2 178 309
  3. Solve for the 41st term of the Fibonacci sequence: 0, 1, 1, 2, 3, 5,...
    • 164 580 141
    • 165 580 141
    • 267 914 296
    • 268 914 296
  4. Who introduced the Fibonacci sequence numbers to the western world through his book Liber Abaci?
    • Leonardo da Vinci
    • Leonardo of Pisa
    • Jacques Philippe Marie Binet
    • Abraham de Moivre
  5. Who discovered the formula in computing the nth term of a Fibonacci sequence?
    • Leonardo da Vinci
    • Leonardo of Pisa
    • Jacques Philippe Marie Binet
    • Abraham de Moivre

Answer Key

  1. 144
  2. 2 178 309
  3. 165 580 141
  4. Leonardo of Pisa
  5. Jacques Philippe Marie Binet

Interpreting Your Score

If you got between 0 and 1 correct answer: You can do it next time. Try it again.

If you got between 2 and 3 correct answers: Maybe you just need more practice.

If you got 4 correct answers: You made it!

If you got 5 correct answers: Perfect!

Books for Math Lovers

Fibonacci Sequence Explained

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