Geometric Sequence

A geometric sequence is a list of numbers where each new number is obtained by multiplying the one before it by the same number every time. We call this the common ratio. This pattern of repeated multiplication creates exponential growth or decay, depending on the ratio, and it can make the numbers increase or decrease, depending on what it is. We must have noticed these kinds of number patterns in real life, like when a population grows, or in science experiments.

A geometric sequence and an arithmetic sequence are both types of number patterns, but they also follow different rules. In an arithmetic sequence, each term is found by adding or subtracting the same number each time. But in a geometric sequence, each term is found by multiplying or dividing by the same non-zero number, called the common ratio.

Geometric Sequence

Definition: A sequence in which every term is set up by multiplying the previous term by a fixed non-zero number (common ratio).

Example: 2, 4, 8, 16, 32 (multiplied by 2 each time)

Formula: a_n​=a₁×r^(n-1)

Growth Type: Exponential (rapid increase or decrease)

Arithmetic Sequence

Definition: A sequence where each term is calculated by adding a common difference to the previous term; the subtraction corresponds to a negative common difference.

Example: 3, 6, 9, 12, 15 (add 3 each time)

Formula: a_n=a+(n−1)×d

Growth Type: Linear (constant increase or decrease)

A geometric sequence is a chain of numbers in which every term is obtained by multiplying the previous term by a fixed number, known as the common ratio.

1. nth Term of a Geometric Sequence
      aⁿ=a₁×r^n-1

a_n= the nth term
‘
a = first term

r = common ratio

n = term number

2. Sum of the First n Terms (Finite Sum)

Sn=a×1-rⁿ/1-r for (r ≠1)

• Sn​ = sum of first n terms

• a = first term

• r = common ratio

• n  = number of terms

3. Sum to Infinity (Infinite Geometric Series)
 

S∞=a1-r for |r|<1

• S∞​ = infinite sum 

• a = first term

• r = common ratio (must be between 1< r < 1)

A geometric sequence is calculated by multiplying the previous one by a fixed number, known as the common ratio. The nth term formula is:

a_n=a₁×r^n-1

Where:

• a_n = the nth term

• a = the first term of the sequence

• r  = the common ratio

• n = the position of the term in the sequence

A recursive formula says that every term in a sequence is based on its preceding term(s). In a geometric sequence, every term is obtained by multiplying the previous term by a fixed number known as the common ratio. The recursive formula is:

a_n=r×a_n-1 For n ≥ 2

Where:

• a_n​ = the nth term

• a_n-1 ​ = the previous term

• r  = the common ratio

• You must also specify the first term: a_1​

Example:

If a₁=2 and r=3, then the sequence is:

2, 6, 18, 54, …

Recursive formula:

• a₁=2
   
• a_n=a_n-1.3 for n>1

The formula for the sum S_n ​of the first n terms of a finite geometric sequence is:

S_n=a₁ a(1-rⁿ)/1-r for r ≠ 1

Where:

• a₁ Is the first term,

• r Is the common ratio between consecutive terms?

• n Is the number of terms to sum?

If the common ratio r=1, the series becomes a constant sequence, and the sum is simply:

S_n=n × a

Derivation

To derive this formula, consider the geometric series:

Sn=a+ar+ar2+ar3+...+arn-1

Multiply both sides by the common ratio (r):

rSn=ar +ar2+ar3+...+arn

Subtract the original series from this new equation:

rSn-Sn= (ar +ar2+ar3+ ...+arn) -(a+ar+ar2+ ... +arn-1)

Simplifying the right-hand side:

(r-1) Sn=arn-a

Solving for Sn 

sn=a(1-rn)1-r for r = 1

Example

Consider a geometric series with the first term a=3, the common ratio r=2, and n=5 the terms.

Using the formula:

Given:
First term a = 4

Common ratio r = 3

Number of terms n = 6

Using the formula:

s6=4(1-36)1-3

First, calculate 36:
36=729
Now, we will substitute into the formula:
s6=4(1-729)1-3=4(-728)-2=2912-2=1456
The answer is 1456

The formula for the sum Sn of the first n terms of a finite geometric series is:
Sn=a1a(1-rn)1-r for r 1

Where:
a  It is the first term of the series.

r It is the common ratio between consecutive terms.
n The number of terms to sum.
If the common ratio r=1, the series becomes a constant sequence, and the sum is simply:
Sn=n a

This formula is derived by multiplying the series by the common ratio and subtracting to eliminate intermediate terms, leaving a simplified expression for the sum.

An infinite geometric series is a sum of infinitely many terms, where each term after the first is found by multiplying the previous term by a common ratio. The sum of such a series is finite only if the absolute value of the common ratio is less than 1.
  S∞=a1-r  for |r|<1
Where:
a It is the first term of the series.
r It is the common ratio between consecutive terms.
∣r∣  Denotes the absolute value of r.

Conditions for Convergence
For the sum of an infinite geometric series to exist, the absolute value of the common ratio must be less than 1:  |r|<1
∣r∣ <1. If ∣r∣ ≥ 1, the series will diverge, meaning it does not have a finite sum. Meaning, it does not have a finite sum.
Example Calculation
Consider the infinite geometric series:
2+1+12+14+18+....
Here:
a=2
r=12
Applying the formula:
S∞=21 - 12=212=4

A geometric sequence model where each term is derived by multiplying the previous one by a constant ratio. This pattern is prevalent in various real-life scenarios, including finance, biology, and technology.

Nature: In a tree branching system, the branches grow in geometric progression, every branch breaks into some small ones it repeats as well.
Architecture: Staircase design measures the heights and widths of the steps in a spiral staircase. In this, we follow the geometric for aesthetic and balance.
Art and Design: In drawing, the sizes decrease as per geometric to create depth and distance in the drawing.
Biology: The population growth of an insect can also be geometric. It doubles every hour.
Technology: Data storage capacity, like memory and drive storage, grows as per the geometric, 16 GB, 32 GB, 64 GB, 128 GB, 25 GB.