Geometric Progression Formulas
In mathematics, a geometric progression(sequence) (also inaccurately known as a geometric series) is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence.
The geometric progression can be written as:
ar^{0}=a, ar^{1}=ar, ar^{2}, ar^{3}, ...
where r ≠ 0, r is the common ratio and a is a scale factor(also the first term).
Examples
A geometric progression with common ratio 2 and scale factor 1 is
1, 2, 4, 8, 16, 32...
A geometric sequence with common ratio 3 and scale factor 4 is
4, 12, 36, 108, 324...
A geometric progression with common ratio 1 and scale factor 5 is
5, 5, 5, 5, 5, 5,...
Formulas
Formula for the nth term can be defined as:
a_{n} = a_{1}⋅r^{n1}
Formula for the common ratio is:
r = 

If the common ratio is:
 Negative, the results will alternate between positive and negative.
Example:
1, 2, 4, 8, 16, 32...  the common ratio is 2 and the first term is 1.
 Greater than 1, there will be exponential growth towards infinity (positive).
Example:
1, 5, 25, 125, 625 ...  the common ratio is 5.
 Less than 1, there will be exponential growth towards infinity (positive and negative).
Example:
1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125 ...  the common ratio is 5.
 Between 1 and 1, there will be exponential decay towards zero.
Example:
4, 2, 1, 0.5, 0.25, 0.125, 0.0625 ...  the common ratio is $\frac{1}{2}$
4, 2, 1, 0.5, 0.25, 0.125, 0.0625 ...  the common ratio is $\frac{1}{2}$.
 Zero, the results will remain at zero.
Example:
4, 0, 0, 0, 0 ...  the common ratio is 0 and the first term is 4.
Geometric Progression Properties
a_{1}⋅a_{n} = a_{2}⋅a_{n1} =...= a_{k}⋅a_{nk+1}
Formula for the sum of the first n numbers of a geometric series
S_{n} =  a_{1}  a_{n}r 1  r  = a_{1}.  1  r^{n} 1  r 
Infinite geometric series where r < 1
If r < 1 then a_{n} > 0,
when n > ∞.
The sum S of such an infinite geometric series is given by the formula:
S =  a_{1} 

Geometric Progression Calculator
Geometric Progression Problems
Problem 1.
Is the sequence 2, 4, 6, 8... a geometric progression?
Solution: No, it is not. (2, 4, 8 is a geometric progression)
Problem 2
If 2, 4, 8... form a geometric progression. What is the 10th term?
Solution: We can use the formula a_{n} = a_{1} ⋅ r^{n1}
a_{10} = 2 ⋅ 2^{101} = 2 ⋅ 512 = 1024
Problem 3
Find the scale factor and the command ratio of a geometric progression if
a_{5}  a_{1} = 15
a_{4}  a_{2} = 6
Solution: there are two geometric progressions. The first one has a
scale factor 1 and common ratio = 2
the second decidion is 16, 1/2
Additional problems:
Geometric progression  problems
Problems involving progressions
Geometric progressions in the math forum
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