Introduction

Geometric Progression

A geometric sequence is a pattern of numbers where you get the next number by multiplying the one before it by the same number every time.

A geometric sequence is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r.

This means if you have one number in the list, just multiply it by a fixed number to find the next one.

The common ratio (r) is the number we use to multiply. You can find it by dividing any number in the list by the one that comes right before it.

The common ratio (r) is obtained by dividing any term by the preceding term.

To find r, just take a term and divide it by the one behind it.

Formula for Common Ratio

r = a₂ / a₁ = a₃ / a₂ = a_n / a_n-1

Where

r = common ratio

a₁ = first term

a₂ = second term

a₃ = third term

a_n-1 = the term before the nth term

a_n = the nth term

Examples of GP

In your math class, we call this Geometric Progression or GP for short.

In mathematics, a geometric progression, also known as a geometric sequence.

It is just a special way numbers follow each other by multiplying.

Look at this example: 4, 16, 64, 256, … This is a GP because if you multiply 4 by 4, you get 16. If you multiply 16 by 4, you get 64. So, the common ratio r is 4.

Look at another one: 10, 5, 2.5, 1.25, … This is also a GP. Here, the numbers are reducing because we are multiplying by 1/2 (or dividing by 2). The common ratio r is 1/2.

The Standard Form

We use a special way to write out any GP using letters.

The general form of a geometric sequence, where r is not equal to 0 is the common ratio and a is a scale factor, equal to the sequence’s start value.

This general form looks like this: a, ar, ar², ar³, ar⁴, and so on.

In this form, 'a' is the first number you start with, and 'r' is what you multiply with to get to the next step.