which statements are true for the given geometric sequence

Guri Bee

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21-10-2024

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Unraveling the Secrets of Geometric Sequences: Identifying True Statements

Geometric sequences, characterized by a constant ratio between consecutive terms, are a fascinating subject in mathematics. Understanding their properties is key to unlocking their secrets and applying them to various real-world situations.

This article delves into the process of identifying true statements about a given geometric sequence. We'll explore common characteristics and use examples to illustrate how to determine their validity.

What is a Geometric Sequence?

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant value called the common ratio.

Example: The sequence 2, 6, 18, 54... is a geometric sequence with a common ratio of 3.

Key Properties of Geometric Sequences:

1. Constant Ratio: The most fundamental property of a geometric sequence is that the ratio between any two consecutive terms is constant. This constant value is called the common ratio.

2. Explicit Formula: We can find any term in a geometric sequence using the explicit formula: a_n = a₁ *r^(n-1), where:

   • a_n is the nth term
   • a₁ is the first term
   • r is the common ratio
   • n is the term number

3. Recursive Formula: Another way to define a geometric sequence is using the recursive formula: a_n = a_(n-1) r, where:

   • a_n is the nth term
   • a_(n-1) is the (n-1)th term
   • r is the common ratio

Determining True Statements about a Given Geometric Sequence

Let's consider the geometric sequence: 4, 8, 16, 32...

1. The common ratio is 2. True. This statement is true because each term is double the previous term (8/4 = 2, 16/8 = 2, 32/16 = 2).

2. The 5th term is 64. True. We can use the explicit formula: a₅ = 4 * 2^(5-1) = 4 * 2⁴ = 64.

3. The sequence is decreasing. False. The sequence is increasing because the common ratio is positive (2). A sequence is decreasing if the common ratio is negative.

4. The sum of the first 5 terms is 124. True. We can calculate the sum using the formula for the sum of a finite geometric series: S_n = a₁(1 - rⁿ) / (1 - r). In this case, S₅ = 4(1 - 2⁵) / (1 - 2) = 124.

Practical Applications:

Understanding geometric sequences is crucial for many applications, including:

• Compound Interest: The growth of money invested at a fixed interest rate follows a geometric pattern.
• Population Growth: The growth of a population can often be modeled using a geometric sequence.
• Radioactive Decay: The decay of radioactive materials follows an exponential pattern, closely related to geometric sequences.

Conclusion:

Identifying true statements about geometric sequences requires understanding their defining properties and utilizing the appropriate formulas. By applying these concepts, we can unravel the secrets of these fascinating mathematical patterns and utilize them in various real-world scenarios.