Binet's formula and thin lens equation

Author: kevinlinxc

README.md

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content/formulas/binets-formula/index.md

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+---
+title: "Binet's Formula"
+description: "The closed form formula for calculating the n-th Fibonacci number."
+summary: "The closed form formula for calculating the n-th Fibonacci number."
+tags: ["math"]
+date: 2025-01-06
+latex: "F_n = \\frac{\\left(\\frac{1+\\sqrt{5}}{2} \\right)^n - \\left(\\frac{1-\\sqrt{5}}{2} \\right)^n}{\\sqrt{5}}"
+---
+
+{{< katex >}}
+
+Binet's formula allows you to calculate the n-th Fibonacci number, \\( \small F_n \\), with a closed form solution. It is as follows:
+
+$$ F_n = \frac{\left(\frac{1+\sqrt{5}}{2} \right)^n - \left(\frac{1-\sqrt{5}}{2} \right)^n}{\sqrt{5}} $$
+
+Where
+
+* \\( \small n \\) is the index of the Fibonacci number you want to calculate. 
+
+## Sources
+
+- [Art of Problem Solving](https://artofproblemsolving.com/wiki/index.php/Binet%27s_Formula)

content/formulas/binets-formula/preview.png

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+---
+title: Thin Lens Equation
+description: "The equation for thin lenses relating focal length, object distance and image distance."
+summary: "The equation for thin lenses relating focal length, object distance and image distance." 
+tags: ["physics", "optics"]
+date: 2025-01-06
+latex: \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}
+---
+
+{{< katex >}}
+
+For light passing near the optical axis of a thin lens, the lens equation relates the focal length of the lens, the object distance, and the image distance. The equation is given by:
+
+$$ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} $$
+
+Where
+
+* \(\small f\) is the focal length of the lens,
+* \(\small d_o\) is the object distance, and
+* \(\small d_i\) is the image distance.
+
+All distances are measured from the lens, and the distances and focal length can be positive or negative. A positive focal length indicates a converging lens, while a negative focal length indicates a diverging lens. If the image distance is positive, the image is real and on the opposite side of the lens from the object. If the image distance is negative, the image is virtual and on the same side of the lens as the object.
+
+## Sources
+
+- [HyperPhysics](http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenseq.html)