Algebra/Geometric Progression (GP)

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2.

Example 1:
A classical example of geometric progression (GP) is population growth. Say the population doubles every 15 years. If the population is 7 billion in 2012, the population in 2027 will be 14 billion.

Since we know that the population doubles ever 15 years, this means from 7 billion in 2012 to 2027 (15 years later) is 14 billion because: $7\cdot 2=14.$

So what will be the population in 2042? If you said 28 billion, you are correct. So how did you get 28? You multiplied 14 by common ratio of 2 to get 28. So to get next term of GP you always multiply the common ratio to current term, and to get previous term you always divide the current term by common ratio.

A more general formula to get the ‘n’th term of a GP is: $a_{n}=a_{1}r^{n-1}$

Where $a_{1}$ is the first term and r is the common ratio of the GP.
So the general terms of GP are:
$a_{1},\;a_{1}r,\;a_{1}r^{2},\;\dots \;a_{1}r^{n-1}$

Finding the sum of ‘n’ terms of an geometric sequence

When needed to find the sum of ‘n’ terms of a GP whose first term is $a_{1}$, you use:
If |r| < 1 then use: $s_{n}=a_{1}{\frac {(1-r^{n})}{(1-r)}}$

If |r| > 1 then use: $s_{n}=a_{1}{\frac {(r^{n}-1)}{(r-1)}}$

Notes

1. The sum of infinite geometric sequence with |r| > 1, is essentially infinite.

1. The sum of infinite geometric sequence with |r| < 1, is $s_{n}={\frac {a_{1}}{(1-r)}}$