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This Geometric Progression Calculator is one of two specialist calculators designed to calculate geometric progression based on specific known criterea. The Geometric Progression First Term Calculator will calculate:

- The sum of the first n-terms of a geometric series when the first term and the common ratio are given.

**Geometric Progression First Term Calculator Parameters:** The number of terms is a natural (counting) number.

Sum of the first n-terms (S_{n}) = |

S_{n} = y_{1}(R^{n - 1})/R - 1S _{n} = (^{ - 1})/ - 1S _{n} = (^{})/S _{n} = × /S _{n} = /S _{n} = |

Geometric Progression First Term Calculator Input Values |

First term of the geometric progression (y_{1}) = |

Common ratio (R) = |

Total number of terms (n) = |

Please note that the formula for each calculation along with detailed calculations is shown further below this page. As you enter the specific factors of each geometric progression first term calculation, the Geometric Progression First Term Calculator will automatically calculate the results and update the formula elements with each element of the geometric progression first term calculation. You can then email or print this geometric progression first term calculation as required for later use.

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The Geometric Progression First Term Calculator has practical application and use in the following fields and disciplines

A geometric sequence (otherwise known as "geometric progression") is a special type of number sequence where the ratio between two consecutive terms is always the same. The general term of a geometrical sequence is denoted by y_{n} (or a_{n}). In this way, we can express a geometrical sequence as y_{1}, y_{2}, y_{3}, , y_{n}. The common ratio in a geometrical sequence is denoted by R.

For example,

2,6,18,54,162,

is a geometric sequence where y_{1} = 2, y_{2} = 6, y_{3} = 18, y_{4} = 54, y_{5} = 162, etc. The common ratio (which can be obtained by dividing any term by its predecessor) here is R = ** a_{n}/a_{n} - 1** = 3. The pattern of this sequence therefore is "multiply by 3."

A geometric series S_{n} represents the sum of the first n terms of a geometric sequence (progression). For example, if we have the following geometric sequence with five terms

1,2,4,8,16

then, the sum of these terms represents the corresponding series S_{n}, where n = 5. Thus,

S_{n} = 1 + 2 + 4 + 8 + 16

=31

=31

The formula of the general term x_{n} of a geometrical sequence (progression) is found in the following way:

First term = y_{1}

Second term = y_{2} = y_{1} ∙ R

Third term = y_{3} = y_{2} ∙ R = y_{1} ∙ R^{2}

Fourth term = y_{4} = y_{3} ∙ R = y_{1} ∙ R^{3}

⋮

General (n - th) term = y_{n} = y_{n - 1} ∙ R = y_{1} ∙ R^{n - 1}

Second term = y

Third term = y

Fourth term = y

⋮

General (n - th) term = y

Now, let's explain how to find the sum of the first n terms in a geometric series. Thus,

S_{1} = y_{1}

S_{2} = y_{1} + y_{2} = y_{1} + R ∙ y_{1}

S_{3} = y_{1} + y_{2} + y_{3} = y_{1} + R ∙ y_{1} + R^{2} ∙ y_{1}

⋮

S_{n} = y_{1} + y_{2} + y_{3} + ⋯ + y_{n} = y_{1} + R ∙ y_{1} + R^{2} ∙ y_{1} + ⋯ + R^{n - 1} ∙ y_{1}

S

S

⋮

S

Multiplying the last expression by the common ratio R yields

R ∙ S_{n} = R ∙ (y_{1} + y_{2} + y_{3} + ⋯ + y_{n} ) = R ∙ y_{1} + R^{2} ∙ y_{1} + R^{3} ∙ y_{1} + ⋯ + R^{n} ∙ y_{1}

Subtracting the previous expression from the last one yields

R ∙ S_{n} - S_{n} = (Ry_{1} + R^{2} y_{1} + R^{3} y_{1} + ⋯ + R^{n} y_{1} ) - (y_{1} + Ry_{1} + R^{2} y_{1} + ⋯ + R^{n - 1}y_{1})

= R^{n} y_{1} - y_{1}

= R

Thus,

S_{n}(R - 1) = y_{1} (R^{n} - 1)

In this way, we obtain the general formula for calculating the sum of the first n terms of a geometric series

S_{n} = *y*_{1}(R^{n} - 1)*/**R - 1*

Multiplying the above formula up and down by -1 gives another version of the sum of the first n terms of a geometric progression (series), i.e.

S_{n} = *y*_{1}(1 - R^{n})*/**1 - R*

This version is used in decreasing geometric sequences, where the common ratio R is smaller than 1.

The following Math tutorials are provided within the Sequences and Series section of our Free Math Tutorials. Each Sequences and Series tutorial includes detailed Sequences and Series formula and example of how to calculate and resolve specific Sequences and Series questions and problems. At the end of each Sequences and Series tutorial you will find Sequences and Series revision questions with a hidden answer that reveal when clicked. This allows you to learn about Sequences and Series and test your knowledge of Math by answering the revision questions on Sequences and Series.