the following sequence has terms that decrease exponentially

Guri Bee

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23-10-2024

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Unraveling the Mystery of Exponentially Decreasing Sequences

Have you ever encountered a sequence where each term is significantly smaller than the one before it? If so, you might be dealing with an exponentially decreasing sequence. These sequences are fascinating for their unique characteristics and applications in various fields.

What Makes a Sequence Exponentially Decreasing?

An exponentially decreasing sequence is characterized by a constant ratio between consecutive terms. This ratio, always less than 1, defines the rate at which the sequence diminishes. As the terms progress, they shrink rapidly towards zero, creating a pattern of exponential decay.

Understanding the Formula:

The general formula for an exponentially decreasing sequence is:

a_n = a_1 * r^(n-1)

Where:

• a_n represents the nth term of the sequence.
• a_1 is the first term.
• r is the common ratio (always less than 1).
• n is the term number.

Examples from the Real World:

Exponentially decreasing sequences find practical application in many real-world scenarios, including:

• Radioactive Decay: The process of radioactive decay follows an exponential pattern, where the amount of radioactive material decreases by half over a specific time period (the half-life).
• Drug Concentration in the Body: After administering a drug, its concentration in the bloodstream decreases exponentially as the body processes and eliminates it.
• Cooling of Objects: The temperature of an object cools down exponentially as it loses heat to its surroundings.

Example: A Geometric Sequence

Let's examine a specific example:

"Consider the sequence 10, 5, 2.5, 1.25, ... What is the common ratio?"

To find the common ratio (r), we simply divide any term by its preceding term. For instance:

r = 5 / 10 = 0.5

This confirms that the sequence is indeed exponentially decreasing, with a common ratio of 0.5.

Key Points to Remember:

• Always look for a constant ratio between consecutive terms.
• The common ratio (r) must be less than 1 for the sequence to be exponentially decreasing.
• The terms approach zero as n increases.

Further Exploration:

For a deeper dive into exponentially decreasing sequences and their applications, consider exploring topics like:

• Geometric Series: The sum of the terms in an exponentially decreasing sequence.
• Half-Life: The time it takes for a substance to decay by half.
• Exponential Growth: The opposite of exponential decay, where terms increase exponentially.

By understanding the principles of exponentially decreasing sequences, you can analyze and interpret data that exhibits exponential decay in a variety of fields.

Attribution:

This article has been inspired by the following GitHub resources:

• [Insert relevant link to GitHub repository or discussion]
• [Insert relevant link to GitHub repository or discussion]

This article has been written to provide an informative and engaging explanation of exponentially decreasing sequences. We hope you found it insightful and beneficial!