Sum of Finite Geometric Series

In geometric sequence is a series by a constant ratio among following expressions. For instance, the series  `(1)/(2)+(1)/(4)+(1)/(8)+...`  is geometric, since each expression except the first know how to be get by multiplying the preceding expression by  `(1)/(2)` . Geometric series are the simplest instances of infinite sequence by finite sums. Historically, geometric sequences play a significant role in the early expansion of calculus, and they maintain to be central in learn of convergence of series.

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Sum of finite geometric series:

A geometric series, also recognized as a geometric sequence, is a series of numbers where each expression after the first is establish by multiplying the preceding one by a fixed non-zero number known the common ratio. For instance, the sequence 3, 9, 27, 81, ... is a geometric series with common ratio 3. Similarly 14, 7, 3.5, 1.75, ... is a geometric series with common proportion 1/2.
           
 The sum of the expressions of a geometric series is recognized as a geometric series. Thus, the general structure of a geometric series is  a,ar,ar²,ar³... and that of a geometric series is a+ar+ar²+ar³+....  where r ≠ 0 is the common ratio and scale factor is a.

Examples for sum of finite geometric series:

Example 1:
 Find the sum of finite n terms and the sum of first 4 terms of the geometric series 2+4+8+16+...............
Solution:
Step 1: the given series is 2+4+8+16+...............
  Step 2: Here, a=2 and r=2 and n=4
Step 3: here r >1
Step 4:     Therefore
s_n = (a(1-r^n))/(1-r)` 
 n = 4

Step 5:          `s_4 = (2(1-2^4))/(1-2)`
Step 6:               `s_4 = (2(1-2^4))/(1-2)`  
Step 7:             `S_4 = 30`

Example 2:
 Find the sum of finite n terms and the sum of first 5 terms of the geometric  series 1+3+5+7+...............

Solution:
Step 1: the given series is  1+3+5+7+..........
Step 2: Here, a=1 and r=2 and n=5

Step 3: here r >1
Step 4:     Therefore
s_n = (a(1-r^n))/(1-r)` 
n = 5



Step 5:          `s_5 = (1(1-2^5))/(1-2)`
Step 6:               `s_5 = (1(1-2^5))/(1-2)`


Step 7:             `S_5 = 31`