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:

$ar^0=a,\ ar^1=ar,\ ar^2,\ ar^3,\ ar^4...$
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 n-th term can be defined as:

$a_n = a_{n-1} \times r$
$a_n = a_1 \times r^{n-1}$

Formula for the common ratio is:

$r = \frac{a_k}{a_{k-1}}$

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

$a^2_k = a_{k-1} \times a_{k+1}$
$a_1 \times a_n = a_{2} \times a_{n-1} = ... = a_k \times a_{n-k+1}$

Geometric Series

The formula for the sum of the first n numbers of a geometric series is:
$a + a_1 + a_2 + ... + a_{n-1}= \frac{ a_1-a_n r}{1-r} = a_1\frac{1-r^n}{1-r}$

or
$a + ar + ar^2 + \cdots + ar^{n-1}= a\frac{1-r^n}{1-r}$

Infinite geometric series where |r| < 1

If |r| < 1 then a_n -> 0, when n -> ∞.
The sum such an infinite geometric series is given by the formula:

$a_1 + a_2 + a_3 + a_4 + \cdots = a_1\frac{1}{1-r}$

or
$a + ar + ar^2 + ar^3 + \cdots = a\frac{1}{1-r}$

which is valid only when |r| < 1.
a₁ is the first term.

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 10-th term?
Solution: We can use the formula a_n = a₁ ⋅ r^n-1
a₁₀ = 2 ⋅ 2^10-1 = 2 ⋅ 512 = 1024

Problem 3
Find the scale factor and the command ratio of a geometric progression if
a₅ - a₁ = 15
a₄ - a₂ = 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

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