Math History question - Fibonacci Proof

In summary, the Fibonacci sequence is a series of numbers where each number is the sum of the two preceding numbers. It was discovered by Italian mathematician Fibonacci and has many applications in mathematics and nature. The sequence can be proven using mathematical induction, first published by French mathematician Édouard Lucas.
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Can someone please help me with the following:

Prove that if the sum of two consecutive intergers is a square than the square of the larger integer will equal the sum of the nonzero squares.

Hint: if n+(n-1) = h^2 then h is odd.

Not really sure where to start.

Thanks in advance
 
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  • #2
Well, $n + (n + 1) = 2n + 1$ is a square by what is given and $(n + 1)^2$ is ... ?
 

FAQ: Math History question - Fibonacci Proof

What is the Fibonacci sequence?

The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding numbers. It starts with 0 and 1, and the sequence continues infinitely as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

Who was Fibonacci?

Fibonacci, also known as Leonardo of Pisa, was an Italian mathematician who lived in the 12th and 13th century. He is best known for introducing the Hindu-Arabic numeral system to Europe and popularizing the use of Arabic numerals in mathematics.

How did Fibonacci come up with his famous sequence?

Fibonacci discovered the sequence while studying the growth of rabbit populations in his book "Liber Abaci". The sequence was already known in Indian mathematics, but Fibonacci brought it to the attention of the Western world.

What is the significance of the Fibonacci sequence in math?

The Fibonacci sequence has many applications in mathematics and nature. For example, it can be used to model the growth of populations, the branching of trees, and the arrangement of seeds in a sunflower. It also appears in many mathematical concepts such as the golden ratio and Lucas numbers.

Is there a proof for the Fibonacci sequence?

Yes, there is a proof for the Fibonacci sequence. It can be proven using mathematical induction, which shows that the sequence follows a recursive rule: F(n) = F(n-1) + F(n-2), where F(0) = 0 and F(1) = 1. This proof was first published by French mathematician Édouard Lucas in the 19th century.

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