# Geometric Progression, Series & Sums

## Introduction

A **geometric sequence** is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the **common ratio** which is denoted by r. The common ratio (r) is obtained by dividing any term by the preceding term, i.e.,

where | r | common ratio |

a_{1} | first term | |

a_{2} | second term | |

a_{3} | third term | |

a_{n-1} | the term before the n th term | |

a_{n} | the n th term |

The **geometric sequence** is sometimes called the **geometric progression** or **GP**, for short.

For example, the sequence 1, 3, 9, 27, 81 is a geometric sequence. Note that after the first term, the next term is obtained by multiplying the preceding element by 3.

The geometric sequence has its sequence formation:

To find the *nth* term of a geometric sequence we use the formula:

where | r | common ratio |

a_{1} | first term | |

a_{n-1} | the term before the n th term | |

n | number of terms |

## Sum of Terms in a Geometric Progression

Finding the sum of terms in a geometric progression is easily obtained by applying the formulas:

*nth* partial sum of a geometric sequence

sum to infinity

where | S_{n} | sum of GP with n terms |

S_{∞} | sum of GP with infinitely many terms | |

a_{1} | the first term | |

r | common ratio | |

n | number of terms |

## Examples of Common Problems to Solve

__Write down a specific term in a Geometric Progression__

**Question**

Write down the 8th term in the Geometric Progression 1, 3, 9, ...

**Answer**

__Finding the number of terms in a Geometric Progression__

**Question**

Find the number of terms in the geometric progression 6, 12, 24, ..., 1536

**Answer**

__Finding the sum of a Geometric Series__

**Question**

Find the sum of each of the geometric series |

**Answer**

__Finding the sum of a Geometric Series to Infinity__

**Question**

**Answer**

__Converting a Recurring Decimal to a Fraction__

Decimals that occurs in *repetition infinitely* or are *repeated in period* are called **recurring decimals**.

For example, 0.22222222... is a recurring decimal because the number 2 is repeated infinitely.

The recurring decimal 0.22222222... can be written as .

Another example is 0.234523452345... is a recurring decimal because the number 2345 is repeated periodically.

Thus, it can be written as or it can also be expressed in fractions.

**Question**

Express as a fraction in their lowest terms.

**Answer**

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