Recursive And Explicit Formula For Geometric Sequence

Let's unravel the fascinating world of geometric sequences, exploring both recursive and explicit formulas, and understanding how each plays a crucial role in defining and predicting the behavior of these sequences. A geometric sequence, at its core, is a sequence where each term is found by multiplying the previous term by a constant value. This constant value is known as the common ratio. The power of geometric sequences lies in their predictable nature, which makes them applicable in various fields such as finance, physics, and computer science.

Understanding Geometric Sequences

A geometric sequence is characterized by a constant ratio between successive terms. This ratio, often denoted as 'r', is the cornerstone of the sequence's behavior.

• Definition: A sequence where each term is obtained by multiplying the previous term by a constant ratio.
• Common Ratio (r): The constant value by which each term is multiplied to get the next term.

Examples of Geometric Sequences

To solidify our understanding, let's look at a few examples:

• 2, 4, 8, 16, 32... (r = 2)
• 1, 3, 9, 27, 81... (r = 3)
• 100, 50, 25, 12.5, 6.25... (r = 0.5)
• 5, -10, 20, -40, 80... (r = -2)

In each of these sequences, dividing any term by its preceding term will always yield the common ratio 'r'. Identifying 'r' is the first step in working with geometric sequences.

Recursive Formula for Geometric Sequences

A recursive formula defines a term in a sequence by relating it to one or more preceding terms. In the context of geometric sequences, the recursive formula expresses the nth term based on the *(n-1)*th term and the common ratio.

Components of a Recursive Formula

• First Term (a₁): The starting point of the sequence. This value must be provided.
• Recursive Relation: The rule that defines how to get the nth term from the *(n-1)*th term.

General Form of a Recursive Formula

The general form of a recursive formula for a geometric sequence is:

• a₁ = [Value of the first term]
• aₙ = r * aₙ₋₁ for n > 1

Where:

• aₙ is the nth term of the sequence.
• aₙ₋₁ is the *(n-1)*th term of the sequence.
• r is the common ratio.

Applying the Recursive Formula: Examples

Let's illustrate the application of the recursive formula with examples.

Example 1:

Consider the geometric sequence: 3, 6, 12, 24, 48...

1. Identify the first term: a₁ = 3
2. Determine the common ratio: r = 6/3 = 2

Therefore, the recursive formula for this sequence is:

• a₁ = 3
• aₙ = 2 * aₙ₋₁ for n > 1

Using this formula, we can find any term in the sequence. For example, to find the 5th term (a₅):

• a₂ = 2 * a₁ = 2 * 3 = 6
• a₃ = 2 * a₂ = 2 * 6 = 12
• a₄ = 2 * a₃ = 2 * 12 = 24
• a₅ = 2 * a₄ = 2 * 24 = 48

Example 2:

Consider the geometric sequence: 10, 5, 2.5, 1.25...

1. Identify the first term: a₁ = 10
2. Determine the common ratio: r = 5/10 = 0.5

Therefore, the recursive formula for this sequence is:

• a₁ = 10
• aₙ = 0.5 * aₙ₋₁ for n > 1

Advantages of the Recursive Formula:

• Simple to understand and implement when you need to find consecutive terms.
• Directly reflects the definition of a geometric sequence.

Disadvantages of the Recursive Formula:

• Inefficient for finding a specific term far down the sequence, as you need to calculate all the preceding terms.
• Not suitable for direct calculation of aₙ without knowing aₙ₋₁.

Explicit Formula for Geometric Sequences

An explicit formula, also known as a general formula, provides a direct way to calculate any term in the sequence without needing to know the previous terms. This is a significant advantage over the recursive formula, especially when dealing with large n values.

Components of an Explicit Formula

• First Term (a₁): The starting point of the sequence.
• Common Ratio (r): The constant value by which each term is multiplied.
• n: The term number you want to find.

General Form of an Explicit Formula

The general form of an explicit formula for a geometric sequence is:

• aₙ = a₁ * r^(n-1)

Where:

• aₙ is the nth term of the sequence.
• a₁ is the first term of the sequence.
• r is the common ratio.
• n is the term number.

Applying the Explicit Formula: Examples

Let's revisit the previous examples and apply the explicit formula.

Example 1:

Consider the geometric sequence: 3, 6, 12, 24, 48...

1. Identify the first term: a₁ = 3
2. Determine the common ratio: r = 2

Therefore, the explicit formula for this sequence is:

• aₙ = 3 * 2^(n-1)

To find the 5th term (a₅) directly:

• a₅ = 3 * 2^(5-1) = 3 * 2⁴ = 3 * 16 = 48

Example 2:

Consider the geometric sequence: 10, 5, 2.5, 1.25...

1. Identify the first term: a₁ = 10
2. Determine the common ratio: r = 0.5

Therefore, the explicit formula for this sequence is:

• aₙ = 10 * (0.5)^(n-1)

To find the 4th term (a₄) directly:

• a₄ = 10 * (0.5)^(4-1) = 10 * (0.5)³ = 10 * 0.125 = 1.25

Advantages of the Explicit Formula:

• Allows direct calculation of any term in the sequence without needing to know the preceding terms.
• Highly efficient for finding terms far down the sequence.

Disadvantages of the Explicit Formula:

• May be less intuitive than the recursive formula for understanding the fundamental nature of a geometric sequence.
• Requires a good understanding of exponents.

Converting Between Recursive and Explicit Formulas

Understanding how to convert between recursive and explicit formulas provides a deeper understanding of geometric sequences and enhances problem-solving skills.

From Recursive to Explicit

Given a recursive formula, we can derive the explicit formula by recognizing the pattern that emerges from the recursive relation.

Steps:

1. Identify a₁ and r from the recursive formula.
2. Substitute a₁ and r into the general explicit formula: aₙ = a₁ * r^(n-1).

Example:

Given the recursive formula:

• a₁ = 5
• aₙ = 3 * aₙ₋₁ for n > 1

1. Identify a₁ = 5 and r = 3.
2. Substitute into the explicit formula: aₙ = 5 * 3^(n-1).

Therefore, the explicit formula is aₙ = 5 * 3^(n-1).

From Explicit to Recursive

Converting from an explicit formula to a recursive formula involves expressing the nth term in terms of the *(n-1)*th term.

Steps:

1. Identify a₁ and r from the explicit formula.
2. Write the recursive relation: aₙ = r * aₙ₋₁

Example:

Given the explicit formula:

• aₙ = 2 * 4^(n-1)

1. Identify a₁ = 2 and r = 4.
2. Write the recursive relation: aₙ = 4 * aₙ₋₁

Therefore, the recursive formula is:

• a₁ = 2
• aₙ = 4 * aₙ₋₁ for n > 1

Applications of Geometric Sequences

Geometric sequences appear in a wide range of real-world applications. Understanding their properties and formulas allows us to model and analyze various phenomena.

Compound Interest

One of the most common applications is in finance, specifically with compound interest. The amount of money accumulated over time with compound interest forms a geometric sequence.

• Formula: A = P (1 + i)^n
  • A = Amount after n years
  • P = Principal amount (initial investment)
  • i = Interest rate per year
  • n = Number of years

In this formula, (1 + i) acts as the common ratio, and the principal amount P is the first term.

Depreciation

Depreciation, the reduction in the value of an asset over time, can also be modeled using geometric sequences. If an asset depreciates at a constant rate, its value each year forms a geometric sequence.

• Formula: Vₙ = V₀ (1 - d)^n
  • Vₙ = Value of the asset after n years
  • V₀ = Initial value of the asset
  • d = Depreciation rate per year
  • n = Number of years

Here, (1 - d) is the common ratio, and the initial value V₀ is the first term.

Population Growth

Under ideal conditions, population growth can sometimes be modeled as a geometric sequence. If a population increases by a constant percentage each year, the population size each year forms a geometric sequence.

• Formula: Pₙ = P₀ (1 + r)^n
  • Pₙ = Population size after n years
  • P₀ = Initial population size
  • r = Growth rate per year
  • n = Number of years

In this case, (1 + r) is the common ratio, and the initial population size P₀ is the first term.

Radioactive Decay

Radioactive decay, the process by which an unstable atomic nucleus loses energy by emitting radiation, follows an exponential decay pattern, which can be related to geometric sequences. The amount of radioactive substance remaining after each half-life forms a geometric sequence.

• Formula: N(t) = N₀ (1/2)^(t/T)
  • N(t) = Amount of substance remaining after time t
  • N₀ = Initial amount of substance
  • t = Time elapsed
  • T = Half-life of the substance

Here, (1/2) is a key component related to the common ratio, and the initial amount N₀ is the first term.

Geometric Series

The sum of the terms in a geometric sequence is called a geometric series. Geometric series have important applications in various areas, including calculating the total amount in an annuity or determining the long-term behavior of certain financial models.

• Formula for the sum of the first n terms: Sₙ = a₁ (1 - r^n) / (1 - r) where r ≠ 1

Computer Science

Geometric sequences and series are also used in computer science, particularly in algorithms and data structures. For example, the analysis of certain algorithms involves geometric series to determine their efficiency.

Practice Problems

To solidify your understanding, let's work through some practice problems.

Problem 1:

Find the 10th term of the geometric sequence with a₁ = 2 and r = 3 using both the recursive and explicit formulas.

Solution:

• Explicit Formula:
  • aₙ = a₁ * r^(n-1)
  • a₁₀ = 2 * 3^(10-1) = 2 * 3⁹ = 2 * 19683 = 39366
• Recursive Formula: (Requires calculating all preceding terms)
  • a₁ = 2
  • a₂ = 3 * 2 = 6
  • a₃ = 3 * 6 = 18
  • a₄ = 3 * 18 = 54
  • a₅ = 3 * 54 = 162
  • a₆ = 3 * 162 = 486
  • a₇ = 3 * 486 = 1458
  • a₈ = 3 * 1458 = 4374
  • a₉ = 3 * 4374 = 13122
  • a₁₀ = 3 * 13122 = 39366

As you can see, the explicit formula is much more efficient for finding a specific term far down the sequence.

Problem 2:

Given the geometric sequence 4, -8, 16, -32..., find the recursive and explicit formulas.

Solution:

1. Identify a₁ = 4 and r = -8/4 = -2

• Recursive Formula:
  • a₁ = 4
  • aₙ = -2 * aₙ₋₁ for n > 1
• Explicit Formula:
  • aₙ = 4 * (-2)^(n-1)

Problem 3:

The initial value of a car is $25,000, and it depreciates at a rate of 15% per year. Find the value of the car after 5 years.

Solution:

1. Identify V₀ = $25,000 and d = 0.15
2. Use the depreciation formula: Vₙ = V₀ (1 - d)^n
3. V₅ = 25000 * (1 - 0.15)⁵ = 25000 * (0.85)⁵ ≈ $11,092.63

Conclusion

Recursive and explicit formulas provide powerful tools for understanding and working with geometric sequences. The recursive formula defines a term based on its predecessor, highlighting the sequential nature of the sequence. The explicit formula, on the other hand, allows for direct calculation of any term, making it highly efficient for finding specific values. By mastering both formulas and understanding their interrelation, you gain a comprehensive understanding of geometric sequences and their applications in various fields.