Math Formula: Geometric Progression

Math Formula: Geometric Progression

Introduction

A geometric progression (GP), also known as a geometric sequence, is a sequence of numbers in which each term is obtained by multiplying the previous term by a fixed number called the common ratio. This concept is foundational in mathematics and appears frequently in finance, computer science, physics, and biology.

The sequence can be written as:

$$ a, ar, ar^2, ar^3, ar^4, \dots $$

Where:

• a is the first term
• r is the common ratio (r ≠ 0)

Geometric Progression Formulas

1. nth Term Formula

To find any term in a geometric progression, use the formula:

$$ T_n = ar^{n-1} $$

This allows us to access any term in the sequence without having to list all the previous terms.

Example 1:

Find the 7th term of the sequence 2, 4, 8, 16, ...

Here, \( a = 2 \), \( r = 2 \), \( n = 7 \)

$$ T_7 = 2 \cdot 2^{6} = 2 \cdot 64 = 128 $$

So, the 7th term is 128.

2. Sum of First n Terms

To find the sum of the first \( n \) terms:

If \( r \neq 1 \), $$ S_n = a \cdot \frac{1 - r^n}{1 - r} $$ or alternatively, $$ S_n = a \cdot \frac{r^n - 1}{r - 1} $$

Example 2:

Calculate the sum of the first 6 terms of the GP: 1, 2, 4, 8, ...

Given: \( a = 1 \), \( r = 2 \), \( n = 6 \)

$$ S_6 = 1 \cdot \frac{2^6 - 1}{2 - 1} = \frac{64 - 1}{1} = 63 $$

3. Sum to Infinity

For converging geometric progressions, where \( |r| < 1 \), the sum to infinity is:

$$ S_{\infty} = \frac{a}{1 - r} $$

This is particularly useful in economics and signal processing, where recurring patterns converge.

Example 3:

Find the sum to infinity of: 10, 5, 2.5, ...

\( a = 10 \), \( r = 0.5 \)

$$ S_{\infty} = \frac{10}{1 - 0.5} = \frac{10}{0.5} = 20 $$

Types of Geometric Progressions

• Increasing GP: When \( r > 1 \), each term increases.
• Decreasing GP: When \( 0 < r < 1 \), each term decreases.
• Alternating GP: When \( r < 0 \), the terms alternate in sign.

Examples:

• Increasing GP: 3, 6, 12, 24 (r = 2)
• Decreasing GP: 100, 50, 25, 12.5 (r = 0.5)
• Alternating GP: 2, -4, 8, -16 (r = -2)

Visualizing Geometric Progressions

Geometric sequences grow or decay exponentially. If plotted on a graph:

• A GP with \( r > 1 \) curves upward steeply.
• A GP with \( 0 < r < 1 \) curves downward toward zero.

Real-World Applications

1. Compound Interest in Banking

Compound interest follows a geometric progression. The amount after n years is:

$$ A = P(1 + r)^n $$

Example: If you invest $1000 at 5% interest for 3 years:

$$ A = 1000(1 + 0.05)^3 = 1000 \cdot 1.157625 = 1157.63 $$

2. Physics: Light Attenuation

The intensity of light passing through a material may decrease geometrically due to absorption, following the equation:

$$ I = I_0 \cdot r^n $$

Where \( I_0 \) is initial intensity and \( r \) is the absorption ratio.

3. Computer Science: Algorithm Complexity

Recursive algorithms like binary search reduce the input space by half each time, forming a geometric sequence of steps:

$$ n, \frac{n}{2}, \frac{n}{4}, \dots, 1 $$

4. Economics: Depreciation of Assets

The value of machinery or electronics often depreciates geometrically over time. For example, a machine might lose 20% of its value yearly:

$$ V = P \cdot (1 - 0.2)^n $$

Geometric Mean and Its Relation to GP

The geometric mean of two numbers \( a \) and \( b \) is:

$$ \text{GM} = \sqrt{ab} $$

In a geometric progression, any middle term is the geometric mean of its adjacent terms:

If \( a, b, c \) are in GP, then: $$ b^2 = ac $$

Example:

Is 6 the geometric mean of 3 and 12?

Check: $$ \sqrt{3 \cdot 12} = \sqrt{36} = 6 $$

Yes, 6 is the geometric mean.

Insertion of Geometric Means

To insert \( n \) geometric means between two numbers \( a \) and \( b \), use:

Let \( a_1 = a \), \( a_{n+2} = b \), and the total terms are \( n+2 \). Then:

$$ r = \left(\frac{b}{a}\right)^{\frac{1}{n+1}} $$

Example:

Insert 2 geometric means between 2 and 16.

Use: $$ r = \left(\frac{16}{2}\right)^{1/3} = 8^{1/3} = 2 $$

So the sequence is: $$ 2, 4, 8, 16 $$

Geometric Progression in Infinite Series

The GP is a building block for infinite series:

Example: Evaluate $$ 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots $$

This is an infinite geometric series where \( a = 1 \), \( r = \frac{1}{2} \)

Sum: $$ S_{\infty} = \frac{1}{1 - \frac{1}{2}} = 2 $$

This explains how infinite repetitions can yield finite results—a principle used in signal processing and game theory.

Practice Exercises

1. Find the 10th term of the GP: 5, 10, 20, 40, ...
2. If the 2nd term is 6 and the 5th term is 162, find \( a \) and \( r \).
3. Determine the sum to infinity of the series: 4, 2, 1, 0.5, ...
4. Insert 3 geometric means between 1 and 16.
5. A machine costing $5000 depreciates 10% annually. Find its value after 5 years.

Conclusion

Geometric progressions are not only fundamental in mathematics but also immensely practical in real-world scenarios. Whether modeling exponential growth, financial investments, or physical decay, the concept of a constant ratio between terms provides a powerful and elegant tool. Understanding geometric progression enhances problem-solving skills and gives insight into natural and technological processes that operate on multiplicative principles.

With formulas for nth term, partial sums, and infinite sums, geometric progression equips learners and professionals alike to analyze and predict complex systems. Its presence in diverse domains—from compound interest to computer algorithms—illustrates its versatility and importance.

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