How to find the common ratio of a geometric sequence

Finding the Common Ratio of a Geometric Sequence

Geometric Sequence

Image by: https://pixabay.com/photos/perspective-symmetry-pattern-5776865/

To find the common ratio of a geometric sequence, you can divide any term in the sequence by the previous term. The common ratio, usually denoted by r, is consistent throughout the sequence.

If the sequence is a1, a2, a3, …, then the common ratio r is given by:

r = a2 / a1 = a3 / a2 = a4 / a3 = …

Example:

Suppose you have the geometric sequence: 3, 6, 12, 24, ….

  1. Take the second term 6 and divide it by the first term 3: 
    r = 6 / 3 = 2
  2. You can check it by dividing the third term 12 by the second term 6: 
    r = 12 / 6 = 2

The common ratio r for this sequence is 2.

Finding the Common Ratio When the Sequence Oscillates Between Positive and Negative

If a geometric sequence oscillates between positive and negative numbers, the process for finding the common ratio is the same. The difference is that the common ratio r will be a negative number. You still divide any term by the previous term, and if the sequence alternates between positive and negative, r will reflect that sign change.

Example:

Consider the sequence: 5, -10, 20, -40, ….

  1. Take the second term -10 and divide it by the first term 5: 
    r = -10 / 5 = -2
  2. To confirm, divide the third term 20 by the second term -10: 
    r = 20 / -10 = -2

The common ratio r is -2, indicating that each term is obtained by multiplying the previous term by -2. This negative ratio causes the sequence to alternate between positive and negative values.

Comments

Post a Comment

Popular posts from this blog

Understanding Heuristics: A Practical Guide to Smarter Problem Solving

Image
Understanding Heuristics: A Practical Guide to Smarter Problem Solving Image by:  https://pixabay.com/illustrations/data-robot-brain-humanoid-mind-7473235/ Introduction In our daily lives, we constantly make decisions, big and small. But how do we tackle complex problems without drowning in endless possibilities? Enter heuristics – a powerful set of mental shortcuts or "rules of thumb" designed to simplify decision-making. Heuristics are especially useful when time is limited, full information is unavailable, or a quick solution is needed. In this post, we'll delve into what heuristics are, how they work, and why they’re indispensable for efficient problem-solving. What Are Heuristics? Heuristics are simplified approaches or techniques that help us make quick, practical decisions. They don’t guarantee the optimal solution but offer a reasonable shortcut to finding workable answers. While heuristics are sometimes viewed as a compromise between accuracy and speed, ...
Read more

The Pythagram Defined

Image
The Pythagram Defined Contributors over a 2.5 Year Period: Tiago Hands ( Final Construction ), Carlos Luna-Mota ( Egyptian Triangles ), Andrzej Kukla ( The Rhombus with Area of 3 ) 06 October 2025 Abstract The Pythagram is a planar geometric structure derived from the 3 : 4 : 5 right triangle and the orthographic projection of a cube. Its construction through defined Cartesian coordinates reveals six interrelated Pythagorean sub-figures that encode proportional symmetry across multiple dimensions. This document formalises the coordinates, connections, and geometric properties of the Pythagram as a reproducible mathematical entity. Coordinate Data Group 1 — Central Core A(16, 18), B(14, 18), C(12, 16), D(12, 14), E(14, 12), F(16, 12), G(18, 14), H(18, 16), I(15, 15), W(15, 17.5), X(12.5, 15), Y(15, 12.5), Z(17.5, 15) Group 2 — Inner Square L(15, 20), M(10, 20), N(10, 15), O(10, 10), P(15, 10), Q(20, 10), R(20...
Read more

The Unicursal Pythagram

Image
🜛 The Unicursal Pythagram 08 October 2025 The unicursal pythagram is a geometric figure drawn on a 6×6 grid . It is made from sixteen points connected by thirty-two straight lines , and the whole pattern can be drawn without lifting the pen . That is what makes it unicursal — it is one continuous path that begins and ends at the same point. The figure is related to the 3:4:5 right triangle , just like the standard pythagram. But while the standard version is about static shapes, the unicursal version adds movement. It shows how all the parts of the pattern connect together in a single, flowing line. Points used (A–P) A(2,2), B(3,4), C(4,2), D(2,3), E(4,4), F(3,2), G(2,4), H(4,3), I(0,3), J(1,5), K(5,5), L(6,3), M(3,0), N(1,1), O(3,6), P(5,1) Drawing order (32 lines) A→B, B→C, C→D, D→E, E→F, F→G, G→H, H→A, A→I, I→J, J→B, B→K, K→L, L→C, C→M, M→N, N→D, D→J, J→O, O→E, E→L, L→P, P→F, F→N, N→I, I→G, G→O, O→K, K→H, H→P, P→M, M→A In simple terms The unicursal pythagram shows...
Read more