SidDash

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)...

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...

Newton-Leibniz Rule Newton-Leibniz's rule helps us to find the derivative of an integral, with respect to some variable, where the limits of integration are some functions of the same variable. While there are many variations of this rule (see note 1 below), we consider here one of the simplest forms. The Theorem: Let ` f(t) ` be a continuous integrable function and let `u(x)` and `v(x)` be some functions of x. Then the derivative of the integral of `f(t)` w.r.t. x is given by,  ` \frac{\text{d}}{\text{d}x}\int_{u(x)}^{v(x)} f(t) dt = f(v(x))\cdot v'(x) - f(u(x)) \cdot u'(x) ` Proof:  The proof for this is very simple. Let g(t) be the anti-derivative of f(t). Then we can write,  ` \int_{u(x)}^{v(x)} f(t) dt = g(t) |_{u(x)}^{v(x)} = g(v(x)) - g(u(x)) ` Now taking the derivative using the chain rule, we get,  ` \frac{\text{d}}{\text{d}x} [g(v(x)) - g(u(x))] = g'(v(x)) \cdot v'(x) - g'(u(x)) \cdot u'(x)   ` However, by our assumption since` g ` is the anti-de...

An Elegant Way to Expand The Inverse Tangent In this post we shall see a method to expand the inverse tangent function ` tan^{-1}(x) ` using an infinite geometric series. The infinite series equips us with a neat and easy to remember technique to find the expansion. A Geometric Series: A series in which a term is obtained by multiplying the previous term by a constant ratio is called a geometric series. This contant ratio is also called the common ratio `r`. For example consider these series: ` 2, 4, 8, 16, 32, ... ` `1, \frac{1}{5}, \frac{1}{25}, \frac{1}{125}, \frac{1}{625}, ... ` In the first series, `r = 2` and in the second series `r = \frac{1}{5}`. Sum of an Infinite Geometric Series: In general, any geometric series can be written as,  `a, ar^{1}, ar^{2}, ar^{3}, ar^{4}, ... ` An infinite series however can only be summed when the series converges i.e. only when `|r| < 1 `. When this is true the sum `S` is given by ` \frac{a}{1-r} ` i.e. ` S = a + ar^{1} + ar^{2} + ar^{3}...