Geometric progression : an insight

Consider the following examples:
(1) In a savings scheme, the amount becomes 5/4 times of itself after every 3 years. The maturity amount (in $) of an investment of $ 800 after 3,6,9 and 12 years will respectively be:
1000, 1250, 1562.5, 1953.125
(2) A bacteria culture doubles itself each hour. So if there is one bacterium at the beginning of the hour, the number of bacteria after 1,2,3,4,5,6 hours can be : 2,4,8,16,32,64.

In both the above examples, we notice a pattern. The succeeding number is a multiple of the preceding number. In other words, if we multiply the preceding number by some fixed number we get the succeeding number. Such a progression is called a geometric progression. The fixed number which when multiplied to the preceding number yields the succeeding number is called the common ratio. This common ratio is usually denoted by ‘r’.

Geometric progression formulas:
Consider the following series:
(1) 3,6,12,24,…..
(2)10,5,2.5,1.25,….
In both the above sequences, we can find the next term by following the pattern.
By definition of the topic we know that it is a list of numbers such that each term is a multiple of the preceding term. Also we just called the fixed multiplying factor the common ratio (r). So if the first term of a G.P is a then the series can be written like this: a, ar, ar^2, ar^3 …..
Following that pattern the geometric progression formula for the nth term is given by:
Tn = ar^(n-1).

Often we also need to find the sum of the terms of such a series. The common ratio r can be r>1, r<1 or r=1. If r>1. The sum of n terms of this depends on the value of this common ratio.
Geometric progression sum formula is given by,
For r > 1, S = a(r^n-1)/(r-1)
For r < 1, S = a(1-r^n)/(1-r)
For r = 1, S = a*n
Where, S = sum of n terms, a = first term and r = common ratio.

Many times, we have a G.P that goes on and on infinitely. The sum of such a progression is not possible if r>1. However if r < 1, then the sum of such a geometric progression exists and it is called the infinite sum. The formula to calculate the infinite sum is as follows:
S = a/(1-r).