Geometric Progression Formulas

In mathematics, a geometric progression (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.

a geometric sequence can be written as:

aq⁰=a, aq¹=aq, aq², q³, ... where q ≠ 0, q is the common ratio and a is a scale factor.

Formulae for the n-th term can be defined as:

a_n = a_n-1.q
a_n = a₁.q^n-1

The common ratio then is:

q =

a_k

a_k-1

A sequence with a common ratio of 2 and a scale factor of 1 is 1, 2, 4, 8, 16, 32...

A sequence with a common ratio of -1 and a scale factor of 3 is 5, -5, 5, -5, 5, -5,...

If the common ratio is:

Negative, the results will alternate between positive and negative.
Greater than 1, there will be exponential growth towards infinity (positive).
Less than -1, there will be exponential growth towards infinity (positive and negative).
Between 1 and -1, there will be exponential decay towards zero.
Zero, the results will remain at zero

Geometric Progression Properties

a²_k = a_k-1.a_k+1
a₁.a_n = a₂.a_n-1 =...= a_k.a_n-k+1

Formula for the sum of the first n numbers of geometric progression

S_n =

a₁ - a_nq 1 - q

= a₁.

1 - qⁿ 1 - q

Infinite geometric series where |q| < 1

If |q| < 1 then a_n -> 0, when n -> ∞ So the sum S of such a infinite geometric progression is:

S =

1 - x

which is valid only for |x| < 1

Geometric Progression Problems

Problem 1) Is the row 2, 4, 6, 8... geometric progression?
Solution: No it is not. (2, 4, 8 is a geometric progression)

Problem 2) If we have 2, 4, 8... is geometric progression. What will be the 10-th term?
Solution: We can use the formula a_n = a₁ . q^n-1
a₁₀ = 2 . 2^10-1 = 2 . 512 = 1024

3) Find out the scale factor and the command ratio of a geometric progression if
a₅ - a₁ = 15
a₄ - a₂ = 6
Solution: there are two geometric progression the first one is with scale factor 1 and common ratio = 2
the second decidion is -16, 1/2

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