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fibonacci.html
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<!DOCTYPE HTML>
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<title>Fibonacci Series</title>
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<h1><a href="index1.html">Golden Ratio</a></h1>
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<h2><a href="#">Relation to the Fibonacci Series</a></h2>
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<a href="#" class="author"><span class="name">Jane Doe</span><img src="images/avatar.jpg"
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<h2 class="featurette-heading">Fibonacci Series <span class="text-muted"></span></h2>
<p class="lead">In mathematics, the Fibonacci numbers, commonly denoted Fn, form a sequence,
called the Fibonacci sequence, such that each number is the sum of the two preceding ones,
starting from 0 and 1. That is,
</p>
<img class="img-fluid" src="images/6.PNG" alt=""><br>
<img class="img-fluid" src="images/7.PNG" alt="">
<p class="lead">for n>0</p>
<p class="lead">For example:</p>
<p class="lead">0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,.......</p>
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<h2 class="featurette-heading">Closed-form expression<span class="text-muted"></span></h2>
<p class="lead">A closed-form expression for the Fibonacci sequence involves the golden ratio:
</p>
<img class="img-fluid" src="images/9.PNG" alt="">
<p class="lead">The golden ratio is the limit of the ratios of successive terms of the Fibonacci
sequence (or any Fibonacci-like sequence), as shown by Kepler:
</p>
<img class="img-fluid" src="images/10.PNG" alt="">
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<h2 class="featurette-heading"><span class="text-muted"></span></h2>
<p class="lead">In other words, if a Fibonacci number is divided by its immediate predecessor in
the sequence, the quotient approximates φ; e.g., 987/610 ≈ 1.6180327868852. These
approximations are alternately lower and higher than φ, and converge to φ as the Fibonacci
numbers increase, and:</p>
<img src="images/11.PNG" alt="">
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