# 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:

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

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

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

#### 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.

a_{1} 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} = a_{1} ⋅ r^{n-1}

a_{10} = 2 ⋅ 2^{10-1} = 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

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