**Introduction**

A sequence is a set of numbers is called as terms, arranged in some particular order. There are two types of sequences; they are Arithmetic sequence and geometric sequence. An arithmetic sequence is a sequence of numbers with the difference between two consecutive terms constant. The difference is called the common difference. A geometric sequence is a sequence of numbers with the ratio between two consecutive terms constant. This ratio is called the common ratio.

**Geometric Sequences Definition:**

In mathematics, a geometric progression is also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54 ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25 ... is a geometric sequence with common ratio 1/2. The sum of the terms of all the terms in geometric sequence is known as a geometric series.

The sum of the terms of a geometric progression is called as a geometric series. Thus, the general form of a geometric sequence is,

** a, ar, ar ^{2}, ar^{3}, ar^{4}……**

and that of a geometric series is,

**a + ar + ar ^{2} + ar^{3} + ar^{4} +…..**

The nth term of a geometric sequences with initial value a and common ratio r is given by,

** a _{n} = ar^{n-1}**

Here, **r → common ratio**

** n → number of term in the series**

Such a geometric sequences also follows the recursive relation,

**a _{n} = ra_{n-1}** for every integer n>=1

Here, **r → Common ratio,**

**n → number of terms in the series**

Generally, to check whether the given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio. The common ratio of a geometric sequence may be negative, resulting in an alternating sequence, with numbers switching from positive to negative and back. For instance,

**1, −3, 9, −27, 81, −243…**

is a geometric sequence with common ratio −3.

Find the 4^{th} partial sum of **a _{n} = 5(2)^{n-1}**

The 4^{th} partial sum of this series is the sum of first four terms,

**a _{1} = 5 (2)^{1-1} => 5*1 => 5**

**a _{2} = 5 (2)^{2-1} => 5*2 => 10**

**a _{3} = 5 (2)^{3-1} => 5*4 => 20**

**a _{4} = 5 (2)^{4-1} => 5*8 => 40**

Therefore, the sum of series is,

** 5 + 10 + 20 + 40 => 75**