Geometric progressions

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Geometric Progressions

Progressions: (lesson 2 of 2)

Definition:

By geometric progression of $m$ terms, we mean a finite sequence of the form

a, \; a r, \; a r^2, \; ..., \; a r^{m-1}

The real number $a$ is known as the first term of the geometric progression, and the real number $r$ is called the ratio of the geometric progression.

Example 1:

Consider the finite sequence of numbers

4, \; 8, \; 16, \; 32, \; 64, \; 128, \; 256, \; 512, \; 1024

In this sequence, the ratio between successive terms is constant and equal to 2.

Here, we have: $a = 4$ and $r = 2$ .

$k-th$ term of the geometric progression:

The $k-th$ term of the geometric progression is equal to

\color{blue}{a r^{k-1}}

Sum of a geometric progression:

The sum of the m terms of a geometric progression is equal to

\color{blue} { S = \left\{ {} \right. }

Example 2:

Consider the geometric sequence $1, \; \frac{1}{2}, \; \frac{1}{4}, \; \frac{1}{8},...$

Here we have: $a = 1$ and $r = \frac {1}{2}$ .

The sum of the first $m$ terms is equal to

\frac{a - a{r^m}}{1 - r} = \frac{1 - 1 \cdot {( \frac{1}{2} )}^m}{1 - \frac{1}{2}} = 2 - {( \frac{1}{2} )}^{m - 1} = 2 - \frac{1}{2^{m - 1}}

This value gets very close to 2 if $m$ is very large.

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