What Is The Limiting Sum Of The Sequence $25, \frac{75}{4}, \frac{225}{16}, \ldots$?

Introduction

The limiting sum of a sequence is a fundamental concept in mathematics, particularly in the field of calculus. It refers to the sum of an infinite series, where each term is a fraction or a decimal value. In this article, we will explore the limiting sum of a specific sequence, which is given by the formula $25, \frac{75}{4}, \frac{225}{16}, \ldots$. This sequence is a geometric progression, where each term is obtained by multiplying the previous term by a fixed constant.

Understanding the Sequence

To find the limiting sum of the sequence, we need to understand the pattern and the formula that generates each term. The given sequence is a geometric progression with a common ratio of $\frac{3}{4}$. This means that each term is obtained by multiplying the previous term by $\frac{3}{4}$. For example, the second term is obtained by multiplying the first term by $\frac{3}{4}$, and the third term is obtained by multiplying the second term by $\frac{3}{4}$.

Formula for the Sequence

The formula for the sequence can be written as:

$$a_n = 25 \left(\frac{3}{4}\right)^{n-1}$$

where $a_n$ is the $n^{th}$ term of the sequence.

Limiting Sum of the Sequence

The limiting sum of the sequence is given by the formula:

$$\sum_{n=1}^{\infty} a_n = \sum_{n=1}^{\infty} 25 \left(\frac{3}{4}\right)^{n-1}$$

This is an infinite geometric series, where each term is a fraction or a decimal value. To find the limiting sum, we need to use the formula for the sum of an infinite geometric series.

Formula for the Sum of an Infinite Geometric Series

The formula for the sum of an infinite geometric series is given by:

$$\sum_{n=1}^{\infty} ar^{n-1} = \frac{a}{1-r}$$

where $a$ is the first term and $r$ is the common ratio.

Applying the Formula

In this case, the first term $a$ is $25$ and the common ratio $r$ is $\frac{3}{4}$. Plugging these values into the formula, we get:

$$\sum_{n=1}^{\infty} 25 \left(\frac{3}{4}\right)^{n-1} = \frac{25}{1-\frac{3}{4}}$$

Simplifying the Expression

To simplify the expression, we need to find a common denominator for the fractions. In this case, the common denominator is $4$. So, we can rewrite the expression as:

$$\frac{25}{1-\frac{3}{4}} = \frac{25}{\frac{4-3}{4}}$$

Further Simplification

To further simplify the expression, we can multiply the numerator and the denominator by $4$. This gives us:

$$\frac{25}{\frac{4-3}{4}} = \frac{25 \times 4}{4-3}$$

Final Simplification

Finally, we can simplify the expression by dividing the numerator by the denominator. This gives us:

$$\frac{25 \times 4}{4-3} = \frac{100}{1}$$

Conclusion

In conclusion, the limiting sum of the sequence $25, \frac{75}{4}, \frac{225}{16}, \ldots$ is $\frac{100}{1}$, which simplifies to $100$. This is the sum of an infinite geometric series, where each term is a fraction or a decimal value.

References

• [1] "Infinite Geometric Series" by Math Open Reference. Retrieved from https://www.mathopenref.com/infinitedoublegeometricseries.html
• [2] "Geometric Progression" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/GeometricProgression.html

Further Reading

• [1] "Calculus: Early Transcendentals" by James Stewart. Cengage Learning, 2016.
• [2] "Mathematics for the Nonmathematician" by Morris Kline. Dover Publications, 2014.

Related Topics

• [1] "Infinite Series" by Math Is Fun. Retrieved from https://www.mathsisfun.com/infinite-series.html
• [2] "Geometric Series" by Khan Academy. Retrieved from https://www.khanacademy.org/math/series-and-series-tests/geometric-series/v/geometric-series
  

Introduction

In our previous article, we explored the limiting sum of a specific sequence, which is given by the formula $25, \frac{75}{4}, \frac{225}{16}, \ldots$. This sequence is a geometric progression, where each term is obtained by multiplying the previous term by a fixed constant. In this article, we will answer some frequently asked questions about the limiting sum of this sequence.

Q: What is the formula for the sequence?

A: The formula for the sequence is given by:

$$a_n = 25 \left(\frac{3}{4}\right)^{n-1}$$

where $a_n$ is the $n^{th}$ term of the sequence.

Q: What is the common ratio of the sequence?

A: The common ratio of the sequence is $\frac{3}{4}$.

Q: How do I find the limiting sum of the sequence?

A: To find the limiting sum of the sequence, you can use the formula for the sum of an infinite geometric series:

$$\sum_{n=1}^{\infty} ar^{n-1} = \frac{a}{1-r}$$

where $a$ is the first term and $r$ is the common ratio.

Q: What is the limiting sum of the sequence?

A: The limiting sum of the sequence is $\frac{100}{1}$, which simplifies to $100$.

Q: Can I use a calculator to find the limiting sum of the sequence?

A: Yes, you can use a calculator to find the limiting sum of the sequence. Simply plug in the values of $a$ and $r$ into the formula for the sum of an infinite geometric series.

Q: What if the common ratio is greater than 1?

A: If the common ratio is greater than 1, the sequence will diverge and the limiting sum will not exist.

Q: What if the common ratio is less than -1?

A: If the common ratio is less than -1, the sequence will diverge and the limiting sum will not exist.

Q: Can I use the formula for the sum of an infinite geometric series to find the limiting sum of any sequence?

A: No, the formula for the sum of an infinite geometric series only applies to sequences that are geometric progressions.

Q: What is a geometric progression?

A: A geometric progression is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed constant.

Q: Can I use the formula for the sum of an infinite geometric series to find the limiting sum of a sequence that is not a geometric progression?

A: No, the formula for the sum of an infinite geometric series only applies to sequences that are geometric progressions.

Q: What if the sequence has a finite number of terms?

A: If the sequence has a finite number of terms, you can use the formula for the sum of a finite geometric series to find the sum of the sequence.

Q: What is the formula for the sum of a finite geometric series?

A: The formula for the sum of a finite geometric series is given by:

$$\sum_{n=1}^{N} ar^{n-1} = \frac{a(1-r^N)}{1-r}$$

where $a$ is the first term, $r$ is the common ratio, and $N$ is the number of terms.

Q: Can I use the formula for the sum of a finite geometric series to find the limiting sum of a sequence?

A: No, the formula for the sum of a finite geometric series only applies to sequences that have a finite number of terms.

Conclusion

In conclusion, the limiting sum of the sequence $25, \frac{75}{4}, \frac{225}{16}, \ldots$ is $\frac{100}{1}$, which simplifies to $100$. We hope that this Q&A article has helped to clarify any questions you may have had about the limiting sum of this sequence.

References

• [1] "Infinite Geometric Series" by Math Open Reference. Retrieved from https://www.mathopenref.com/infinitedoublegeometricseries.html
• [2] "Geometric Progression" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/GeometricProgression.html

Further Reading

• [1] "Calculus: Early Transcendentals" by James Stewart. Cengage Learning, 2016.
• [2] "Mathematics for the Nonmathematician" by Morris Kline. Dover Publications, 2014.

Related Topics

• [1] "Infinite Series" by Math Is Fun. Retrieved from https://www.mathsisfun.com/infinite-series.html
• [2] "Geometric Series" by Khan Academy. Retrieved from https://www.khanacademy.org/math/series-and-series-tests/geometric-series/v/geometric-series