What are Number Series?

Number series, also known as sequences, are numbers arranged in a specific order or pattern. Each number in the series is derived based on a rule or pattern, which may involve arithmetic operations, geometric progression, or other mathematical relationships.
In other words, number series are functions where domain X is an order and codomain Y is a value corresponding to the order.

Fibonacci sequence

Let's consider Fibonacci sequence as an example. It is a well-known numerical sequence in mathematics that starts with 0 and 1, and each subsequent number is the sum of the two preceding numbers:


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# Define function to generate Fibonacci sequence
def generate_fibonacci_sequence(n):
    # Initialize the sequence with the first two Fibonacci numbers
    sequence = [0, 1]
    
    # Generate Fibonacci sequence up to the nth number
    for i in range(2, n):
        # Calculate the next Fibonacci number by adding the last two numbers
        next_number = sequence[i-1] + sequence[i-2]
        
        # Append the next Fibonacci number to the sequence
        sequence.append(next_number)
    
    # Return the generated Fibonacci sequence
    return sequence

# Number of Fibonacci numbers to generate
n = 10

# Generate Fibonacci sequence
fib_seq = generate_fibonacci_sequence(n)

# Print the generated Fibonacci sequence
print(f'Fibonacci sequence with {n} elements is {fib_seq}')

The Fibonacci sequence finds numerous applications in various real-life tasks and scenarios. Some of the common areas where the Fibonacci sequence is used include:

Financial Planning: In finance, the Fibonacci sequence is used in technical analysis to predict potential support and resistance levels in the stock market. Traders and analysts use Fibonacci retracements and extensions to identify possible price levels where a financial instrument may reverse or continue its trend;
Computer Algorithms: The Fibonacci sequence plays a role in several computer algorithms and data structures. For example, it is used in dynamic programming algorithms, recursive functions, and in generating Fibonacci heaps, which are data structures used for priority queue operations;
Computer Graphics: In computer graphics, the Fibonacci sequence can be utilized to generate visually appealing patterns and designs, especially in fractals and recursive algorithms.

Arithmetic sequence

Another popular sequence is an arithmetic sequence, also known as an arithmetic progression, a sequence of numbers in which the difference between consecutive terms is constant.


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# Function that generates arithmetic progression
def generate_arithmetic_progression(a, d, n):
    # Initialize an empty list to store the arithmetic progression
    progression = []
    
    # Initialize the current term to the first term of the progression
    current_term = a
    
    # Generate the arithmetic progression up to the nth term
    for i in range(n):
        # Append the current term to the progression list
        progression.append(current_term)
        
        # Calculate the next term by adding the common difference to the current term
        current_term += d
    
    # Return the generated arithmetic progression
    return progression

# Example usage
a = 2  # First term
d = 3  # Common difference
n = 10  # Number of terms to generate

# Generate arithmetic progression
progression = generate_arithmetic_progression(a, d, n)

# Print the generated arithmetic progression
print(f'Arithmetic sequence with {n} elements is {progression}')

Arithmetic sequences can also be used in various real-life tasks and applications. Some examples include:

Financial Planning: In finance, arithmetic sequences are used to model and analyze various financial scenarios, such as loan payments, mortgage payments, and investments with fixed or changing amounts over time;
Project Management: In project management, arithmetic sequences can be used to track and predict project progress, resource allocation, and budget planning;
Time Series Analysis: In time series analysis, arithmetic sequences can be used to model and forecast data that exhibit a consistent linear trend over time.

Geometric sequence

Finally, let's consider a geometric sequence, also known as a geometric progression. It is a sequence of numbers in which each term is obtained by multiplying the previous term by a fixed, non-zero value called the common ratio (denoted by 'r').


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# Function that generates geometric progression
def generate_geometric_progression(a, r, n):
    # Initialize an empty list to store the geometric progression
    progression = []
    
    # Initialize the current term to the first term of the progression
    current_term = a
    
    # Generate the geometric progression up to the nth term
    for i in range(n):
        # Append the current term to the progression list
        progression.append(current_term)
        
        # Calculate the next term by multiplying the current term by the common ratio
        current_term *= r
    
    # Return the generated geometric progression
    return progression

# Example usage
a = 3  # First term
r = 2  # Common ratio
n = 10  # Number of terms to generate

# Generate geometric progression
progression = generate_geometric_progression(a, r, n)

# Print the generated geometric progression
print(f'Geometric sequence with {n} elements is {progression}')

Geometric sequences can be used to solve the following real-life tasks:

Financial calculations: In finance, geometric sequences are used for compound interest calculations. When interest is compounded over time, an investment's amount grows according to a geometric sequence;
Population growth: In biology and ecology, geometric sequences can be used to model the growth of populations in which each generation produces a fixed multiple of the previous one;
Exponential decay: Geometric sequences can also represent exponential decay processes, such as the decay of radioactive isotopes or the decrease in the concentration of a substance over time;
Pricing and discounting: In business and economics, geometric sequences can be used to model pricing strategies and discounting schemes.

Note

There are many other sequences in addition to those described above. The most common task is to determine the pattern by which the sequence elements are calculated. Knowing this pattern, we can conduct a more detailed analysis of each sequence.

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Section 1. Chapter 2