Geometrical series is taken highly important when preparing for competitive exams like SBI, PNB, clerk etc. So, what is a Geometrical Series exactly? This is series formed by the multiplying the first term by a number to get the another and the process will be continued to make a number series that will increase swiftly and given as the name Geometric Progression.

The sum of geometric series would be finite as long as long the value of the ratio is less than one or a number close to zero. When a number comes closer to zero, it becomes infinitely small, allowing a sum to be calculated for the series containing infinitely small numbers. The sum is calculated for the self-similar series.

\[\ s_n = \frac{a(1-r^n)}{1-r}\]

The terms of geometric series mean ratio of successive terms in a series is always a constant. You just need to understand the relationship between two terms i.e. a and r. Where a is the first term in the series and r is the common ratio. The behavior of r depends on many terms in mathematics –

- If the value of r lies between -1 and +1 then terms in the series will get smaller and it will reach to the zero in the limit soon. In this case, the value of r is one-half and the sum of the series is one.
- If the value of r is greater than one less than -1 then series would get larger in size and magnitude.
- The value of r is one in the series then all terms should be equal in the series.
- If the value of r is -1 then the sum of terms will lie between two values always.

A Geometric sequence is a sequence where each successive term is formed by multiplying the previous one with a certain number. The other name for the Geometric sequence is Geometric progression or GP in mathematics. Here, r is the common ration and a1, a2, a3 and so on are the different terms in the series. The formula for GP is given as below –

\[\large g_{n}=g_{1}\;r^{n-1}\]

Where,

g_{n} – n^{th} term that has to be found

g_{1} – 1^{st} term in the series

r – common ratio

The geometrical distribution presents the number of failures before you succeed in a series of Bernoulli trials. The Geometric distribution formula in mathematics is given by the density function as mentioned below –

Take an example, where you are asking to the people outside a polling booth who they voted. Few would reply the truth while others may simply confuse you with wrong answers. So, there are chances of failure before you actually succeed in your observation.

The sum of an infinite GP forms an infinite geometric series in mathematics and there is no last term for this series. The general form of an infinite Geometric series is given as –

\[\large a_{1}+a_{1}r+a_{1}r^{2}+a_{1}r^{3}+….+a_{1}r^{n-1}\]

The formula for the resultant sum of the Infinite Geometric Series is,

\[\large S_{\infty}=\frac{a_{1}}{1-r};\left|r\right|<1\]

where a1 is the first term and r would be the common ratio. This is possible to calculate the sum of infinite GP as well but you will not get a final answer.

Common ratio, r |
Start term, a |
Example series |
---|---|---|

10 | 4 | 4 + 40 + 400 + 4000 + 40,000 + ··· |

1/3 | 9 | 9 + 3 + 1 + 1/3 + 1/9 + ··· |

1/10 | 7 | 7 + 0.7 + 0.07 + 0.007 + 0.0007 + ··· |

1 | 3 | 3 + 3 + 3 + 3 + 3 + ··· |

−1/2 | 1 | 1 − 1/2 + 1/4 − 1/8 + 1/16 − 1/32 + ··· |

–1 | 3 | 3 − 3 + 3 − 3 + 3 − ··· |