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It appears in any system where the growth (or decrease), at any stage is dependent on the level reached.
Common examples:
Compound interest (surely, nobody, except elementary maths pupils, uses simple interest!).
Depreciation.
Radioactive decay.
Also,
Pyramid selling schemes - until they hit the buffers of saturation!
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Why is geometric progression preferred over arithmetic progression?
The question cannot be answered because it assumes something which is simply not true. There are some situations in which arithmetic progression is more appropriate and others in which geometric progression is more appropriate. Neither of them is "preferred".
What is the common ratio of the geometric progression?
The common ratio is the ratio of the nth term (n > 1) to the (n-1)th term. For the progression to be geometric, this ratio must be a non-zero constant.
Application of geometric progression in business?
Immediately springing to mind, geometric progression is used in accountancy in finding the Net Present Value of projects (specifically, the value of money each year based on the discount factor). It is also used in annuities, working out monthly repayments of loans and values of investments - compound interest is a geometric progression.
Insert 3 geometric means between 1 and 256?
Geometric progression 1, 4, 16, 64, 256 would seem to fit...
How do you find the ratio in the geometric progression?
Divide any term, except the first, by the term before it.
