Geometric Bites

Arithmetic and Geometric Sequences Arithmetic and geometric sequences are two fundamental types of numerical progressions. They describe how quantities grow or shrink by addition or by multiplication, and they form the foundation for topics such as series, summation formulas, and exponential growth. 1. Arithmetic Sequence An arithmetic sequence is a list of numbers in which each term differs from the previous one by a fixed amount called the common difference d . a, a + d, a + 2d, a + 3d, … , a + (n − 1)d a – first term d – common difference The n th term, denoted T n , is given by: T n = a + (n − 1)d Each new term is obtained by adding d to the previous term. The difference between consecutive terms remains constant: T k+1 − T k = d Example: If a = 4 and d = 3, the sequence is 4, 7, 10, 13, 16, … The 20th term is T 20 = 4 + (20 − 1)×3 = 61. 2. Geometric Sequence A geometric sequence is a list of numbers where each term is fo...

The Trapezoidal Rule — A Visual, First-Principles Introduction 11 November 2025 · @mathematics.proofs To understand integration deeply, it helps to think geometrically. Instead of memorising formulas, we begin by observing shapes. The goal is simple: break the interval into small pieces, estimate the area on each piece, and add everything together. Rectangles give a basic approximation. But functions rarely behave perfectly flat on every interval. A better idea is to allow the top edge to tilt. This leads us naturally to trapezoids . 1) Partitioning the Interval Consider an interval from a to b . We divide it into n equal parts. Δx = (b − a) / n x i = a + i·Δx for i = 0, 1, 2, …, n At each x i , we record the height of the function f(x i ). These sample values will guide our area estimates. 2) One Slice: Rectangle + Triangle Focus on a single subinterval [x i , x i+1 ]. If we draw a vertical line at x i and take a height of f(x i ), we obtain a rectangle o...

Function Composition: A Simple Way to Organise Your Mathematics Function composition is one of the most useful tools in mathematics. It allows us to combine several steps into a single process, keeping our work neat, organised, and easy to reuse. Rather than performing one operation after another by hand, composition lets us build those steps into a single function. Once you become comfortable with function composition, you never want to go back to doing everything manually. It reduces clutter, helps you work systematically, and allows you to achieve remarkable results with only a few lines of equations. What Is Function Composition? A function takes an input, performs an operation, and produces an output. Function composition takes this idea further: it links functions together so that the output of one becomes the input of the next. We write this as: (f ∘ g)(x) = f(g(x)) This means that g acts first, then f . The circle symbol ∘ simply means “do one funct...

Common Transformations in Geometry: A Beginner’s Guide In geometry, a transformation is a rule that changes the position or appearance of a shape or a set of points. Some transformations simply move a shape to a new location, while others may turn it, resize it, or reflect it. Understanding these ideas helps us describe movement and change in a clear mathematical way. This guide introduces the most common transformations: translation, rotation, scaling, reflection, shear, and projection. Each section includes a simple example to make the ideas easier to follow. 1) Translation — Moving A translation shifts every point of a shape by the same amount. Nothing about the shape itself changes — not its size, not its proportions, and not its orientation. Only its position is different. Example: Imagine a triangle on graph paper. If every point of the triangle moves 3 units to the right and 2 units up, the triangle looks exactly the same — it simply appears somewhere else on th...