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**Geometric progression series**

A geometric progression series is one in which any two consecutive terms have the same ratio. As a result, we can find the subsequent term by multiplying the common ratio by the previous term.

This is how the series looks:Â Â **a, ar, ar ^{2}, ar^{3}, ar^{4}, . . . . .**

where common ratio(r)=2nd term/1st term (T2/T1) or (T3/T2)

Standard Formula to find the sum of series in G.P = Â **a(1 â€“ r ^{n})/(1 â€“ r)**

Given First term(a), common ratio(r), Nth term(total number of terms ) in Series, The task is to find Sum of the Geometric Progression Series.

Example: 2,6,18,54,162,486,1458,. . . . . . . .

Here a=2 , r=6/2 =3 , let n=10

Formula to find Nth term =**Â ar ^{n}^{–}^{1Â Â }**

= 39366

sum of series = **a(1 â€“ r ^{n})/(1 â€“ r) **=59048

**Examples:**

**Example1:**

**Input:**

Given Total number of terms=4 Given First Term = 3 Given common ratio = 3

**Output:**

The sum of the given geometric progression series = 120

**Example 2:**

**Input:**

Given Total number of terms=6 Given First Term = 4 Given common ratio = 5

**Output:**

The sum of the given geometric progression series = 15624

## Program to Find Sum of Geometric Progression Series

Below are the ways to find the Sum of the Geometric Progression Series.

### Method #1: Using Mathematical Formula (Static Input)

**Approach:**

- Import math module using the import keyword.
- Give the Total number of terms as static input and store it in a variable.
- Give the first term as static input and store it in another variable.
- Give the Common Ratio as static input and store it in another variable.
- Calculate the given Sum of Geometric Progression Series by using the standard mathematical formula
**a(1 â€“ r**and store it in a variable.^{n})/(1 â€“ r) - Print the sum of the Geometric Progression series.
- The Exit of the program.

**Below is the implementation:**

# Import math module using import keyword. import math # Give the Total number of terms as static input and store it in a variable. tot_trms = 10 # Give the first term as static input and store it in a variable. fst_trm = 2 # Give the Common Ratio as static input and store it in a variable. commn_diff = 3 # Calculate the given Sum of Geometric Progression Series by using standard mathematical formula # a(1 â€“ r**n)/(1 â€“ r) and store it in a variable. sum_geoprog = (fst_trm*(1-(commn_diff)**tot_trms))//(1-commn_diff) # Print the sum of Geometric Progression series . print("The sum of the given geometric progression series = ", sum_geoprog)

**Output:**

The sum of the given geometric progression series = 59048

### Method #2: Using Mathematical Formula (User Input)

**Approach:**

- Import math module using the import keyword.
- Give the Total number of terms as User input using the int(input()) function and store it in a variable.
- Give the first term as User input using the int(input()) function and store it in another variable.
- Give the Common Ratio as User input using the int(input()) function and store it in another variable.
- Calculate the Sum of Geometric Progression Series by using the standard mathematical formula
**a(1 â€“ r**and store it in a variable.^{n})/(1 â€“ r) - Print the sum of the Geometric Progression series.
- The Exit of the program.

**Below is the implementation:**

#Import math module using the import keyword. import math #Give the Total number of terms as User input usingÂ theÂ int(input())Â functionÂ and store it in a variable. tot_trms = int(input("Given Total number of terms =")) #Give the first term as User input using the int(input()) function and store it in another variable. fst_trm = int(input("Given First Term = ")) #Give the Common Ratio as User input usingÂ theÂ int(input())Â functionÂ and store it in another variable. commn_diff = int(input("Given common ratio = ")) #Calculate the Sum of Geometric Progression Series by using standard mathematical formula #a(1 â€“ r**n)/(1 â€“ r) and store it in a variable. sum_geoprog = (fst_trm*(1-(commn_diff)**tot_trms))//(1-commn_diff) #Print the sum of Geometric Progression series . print("The sum of the given geometric progression series = ", sum_geoprog)

**Output:**

Given Total number of terms =6 Given First Term = 4 Given common ratio = 5 The sum of the given geometric progression series = 15624

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