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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 Two Terms Calculator will calculate:

- The sum of the first n-terms of a geometric series when any two terms of the series are given.

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

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

S_{n} = y_{p} ∙ y_{q}/y_{p}^{n/ q - p} - 1/y_{q}/y_{p}^{p - 1/q - p} ∙ y_{q}/y_{p}^{1/q - p} - 1S _{n} = ∙ /^{/ - } - 1//^{ - 1/ - } ∙ /^{1/ - } - 1S _{n} = ∙ ^{/ } - 1/^{/} ∙ ^{1/} - 1S _{n} = ∙ ^{} - 1/^{/} ∙ ^{1/} - 1S _{n} = ∙ - 1/^{} ∙ ^{} - 1S _{n} = ∙ / ∙ - 1S _{n} = / ∙ S _{n} = /S _{n} = |

Geometric Progression Two Terms Calculator Input Values |

The p^{th} term of the geometric progression (y_{p}) = |

The q^{th} term of the arithmetic progression, (y_{q}) = |

The position of the p^{th} term in the sequence, (p) = |

The position of the q^{th} term in the sequence (q) = |

The number of terms for which the sum is calculated (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 two terms calculation, the Geometric Progression Two Terms Calculator will automatically calculate the results and update the formula elements with each element of the geometric progression two terms calculation. You can then email or print this geometric progression two terms calculation as required for later use.

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The Geometric Progression Two Terms Calculator has practical application and use in the following fields and disciplines

For a detailed theoretical description on the calculations and comprehension of this Geometric Progression Calculator please refer to the supporting documentation in our Geometric Progression Calculator for sum of the first n-terms of a geometric series when the first term and the common ratio are given..

If you know two terms of a geometrical progression (and obviously the position they occupy in the sequence), you can calculate the sum of as many terms as you want just by inserting the two terms yp and yq (p < q), the positions p and q they occupy in the sequence (i.e. which terms are yp and yq in the sequence) and the number n of the terms you want to include in the sum calculation. Then, with just a click, you will obtain the desired result.

For example, in the geometrical sequence 2, 6, 18, 54, 162, 486, suppose you are given only y2 = 6 and y5 = 162 and nothing else (p = 2 and q = 5). The exercise may require finding the sum of the first 6 terms (n = 6). Using the formula given in the calculator, we obtain:

S_{n} = *y*_{p} ∙ *y*_{q}*/**y*_{p}^{n/ q - p} - 1*/**y*_{q}*/**y*_{p}^{p - 1/q - p} ∙ *y*_{q}*/**y*_{p}^{1/q - p} - 1

S_{6} = *6 ∙ **162**/**6*^{6/5 - 2} - 1*/**162**/**6*^{2 - 1/5 - 2} ∙ *162**/**6*^{1/5 - 2} - 1

=*6 ∙ *27^{6/3} - 1*/**27*^{1/3} ∙ 27^{1/3} - 1

=*6 ∙ [27*^{2} - 1])*/**3 ∙ [3 - 1]*

=*6 ∙ [729 - 1]**/**3 ∙ 2*

= 728

S

=

=

=

= 728

Proof:

2 + 6 + 18 + 54 + 162 + 486

= 8 + 18 + 54 + 162 + 486

= 26 + 54 + 162 + 486

= 80 + 162 + 486

= 242 + 486

= 728

= 8 + 18 + 54 + 162 + 486

= 26 + 54 + 162 + 486

= 80 + 162 + 486

= 242 + 486

= 728

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.