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A Theorem on Arithmetic Progression and Divisibility In this post, we briefly discuss a particular theorem concerning the divisibility of a sequence of terms in arithmetic progression. Before that, let's take a quick review of an arithmetic progression.  Arithmetic Progression: A sequence of numbers that have the same common difference between consecutive terms is said to be an arithmetic progression. For example,  ` 1, 5, 9, 15, 19, ... `  is a sequence of numbers that have a common difference of `4`. Mark that is not the absolute difference but rather algebraic difference. Thus the above is said to be an increasing AP with common difference `4`.  The common difference is taken algebraically as ( any term) - (the previous term). More examples: ` \frac{1}{2} , 1, \frac{3}{2}, 2, \frac{5}{2}, ... ` ` \sqrt{10}, \sqrt{10} - \sqrt{2}, \sqrt{10} - 2\sqrt{2}, ... ` See also that any series of consecutive terms taken from an arithmetic progression also form an arithmetic ...

Derivative of Product and Quotient The derivatives are widely used throughout Mathematics and the Sciences, and the Identities simplifying computations are invaluable tools. In this post, our intent is to derive the multiplication and division identities of derivatives using the very first principles. Derivative Suppose there be a continuous real function ` y = f(x) ` defined over an open interval `(a, b) \subseteq \mathbb{R}`. Then the derivative of the function `f(x)` at some point `c \in (a, b)` represented by, `f'(c) or \frac{d}{dx}f(x)` at `x = c` gives the rate of change of the function at the point `c` with respect to the independent variable. This idea however is still vague and we need a precise mathematical definition.   Right-hand & Left-hand Rate: In the neighbour of `c`, let us define an interval `I = [c - h, c + h]` for some ` h > 0`. Then we may define the right-hand average rate of change of the function in the interval as,  `\frac{f(c+h) - f(c)...

A Guidebook To Mechanism In Organic Chemistry (Descont.) Exciting News! ✨ I am thrilled to share that I am adding a new book to the DesCont series:    📘A Guidebook to Mechanism in Organic Chemistry by Peter Sykes This will also be the first impression for a theme in Chemistry in this blog. If you have followed the previous writings under the DesCont series, you now that they are meant to clarify, extend & reflect ideas from the book being explored. I hope this new addition brings you insights, clarity, and a good time reading along. 🌿 The book is now available in the DesCont page. Fig 1: A methane molecule (one of the simplest organic compounds)

Infinitely Many Primes?! Prime numbers are one of the most fascinating aspects of Number Theory. Despite being one of the simplest to define, they have amazed mathematicians for centuries. In the post, we see, how one of the earliest theorems on prime numbers was proved by Euclid about 2000 years ago! Prime Numbers:  A natural number greater than 1, that is divisible only by 1 and itself is called a prime number. In other words, if ` p \in N`, `p > 1`  has only `1` and `p` as its factor, then the number `p` is called a prime number. For example `19` has no factors except `1` and itself. Thus it's a prime number. The first `20` prime numbers are listed below:  ` 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29` `31, 37, 41, 43, 47, 53, 59, 61, 67, 71` Fig 1: The distribution of first 20 prime numbers (In x-axis, 10 units represent the gap between two primes) So Many Primes! It's then exciting to know certain properties of these special numbers, isn't it? Let's then see a few no...

(Refresh the browser once more for the equations to load properly) Transient States in RL Circuits: The Journey to Steady-State  In this article, we attempt to obtain the complete solution of an RL circuit, including the transient state. We first consider the case of a DC voltage, where the current gradually builds up and after that, we take up the special case of AC Voltage which is often skipped in elementary courses. Fig: Growth of current in cases of DC and AC voltages DC Voltage: A DC Voltage refers to a steady voltage that does not vary with time. The below diagram shows a simple DC circuit involving a resistor `R` and an inductor `L`. A cell acts as the source of a DC emf E. Fig: A simple RL circuit with DC emf From Kirchhoff's Voltage Law, we know that the magnitude of the net Voltage drop across a loop is equal to the magnitude of the net emf in that loop. The voltage drop due to the resistor is `iR` and due to the inductor is `L \frac{di}{dt}`. Thus we m...

SHM Phasors This post is only an extension to the previous article on AC Phasors. The reader is advised to read the article on AC Phasors here  first, to gain an elementary idea on phasors. In this article, we consider phasors used in describing Simple Harmonic Motion. Simple Harmonic Motion: The motion governed by a force proportional to the displacement of the particle from the mean position and directed towards the mean position is called simple harmonic motion. i.e. for an ideal particle of mass `m`, we have, ` F = -kx` i.e.   ` m \frac{\text{d}^{2}x}{\text{dt}^{2}} = -kx ` Suppose we have the mass particle `m` attached to a spring that applies force according to the above equation. We stretch it from the mean position to some distance say `a` and then release it.  The differential equation of this motion is then, ` \frac{\text{d}^{2}x}{\text{dt}^{2}} = - \frac{k}{m}x` Since both `k` and `m` are positive quantities, we can let `\frac{k}{m} = w^{2}`, for some `w...

(Refresh the browser once more for the equations to load properly) Animating AC Phasors In this post, we try to get an intuitive feeling about special quantities called phasors, used to represent current and voltage in AC circuits. We will see the theory at work, learning about complex numbers, sinors, and how these are used to represent voltage and current. At the end, we'll be able to input our custom values in the animation! Sinusoids: Any quantity that varies in the form of a sine or a cosine function is called a sinusoid. When a sinusoidally time-varying voltage is applied across an electrical component we get a corresponding sinusoidally time-varying current which we call an Alternating Current (AC). The voltage applied is called the alternating voltage. In the general form if ` v(t) = V_{0} sin(ωt) `, then the current is in the form ` i(t) = I_{0} sin(ωt + φ) `, where ` φ ` is called the phase difference, created by the properties of the ele...