{ \frac 1}{10}\left(1.1^{50 / 10}+1.1^{51 / 10}+1.1^{52 / 10}+\ldots+1.1^{99 / 10}\right)$}$ Is An Approximation For A. { \int_5^{9.9 1.1^x , Dx$}$B. { \int_5^{10} 1.1^x , Dx$} C . \[ C. \[ C . \[ \frac{1}{10} \int_{50}^{99}

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Introduction

In mathematics, approximating integrals is a crucial concept that helps us understand the behavior of functions and their areas under curves. One of the most effective ways to approximate integrals is by using geometric series. In this article, we will explore how to approximate a specific integral using a geometric series.

The Geometric Series Formula

A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The formula for the sum of a geometric series is given by:

Sn=a(1−rn)1−rS_n = \frac{a(1 - r^n)}{1 - r}

where aa is the first term, rr is the common ratio, and nn is the number of terms.

The Given Integral

The given integral is:

110(1.150/10+1.151/10+1.152/10+…+1.199/10)\frac{1}{10}\left(1.1^{50 / 10}+1.1^{51 / 10}+1.1^{52 / 10}+\ldots+1.1^{99 / 10}\right)

This integral can be rewritten as a geometric series with a common ratio of 1.11.1 and a first term of 1.150/101.1^{50/10}.

Approximating the Integral

To approximate the integral, we can use the formula for the sum of a geometric series:

Sn=a(1−rn)1−rS_n = \frac{a(1 - r^n)}{1 - r}

In this case, a=1.150/10a = 1.1^{50/10}, r=1.1r = 1.1, and n=50n = 50. Plugging these values into the formula, we get:

S50=1.150/10(1−1.150)1−1.1S_{50} = \frac{1.1^{50/10}(1 - 1.1^{50})}{1 - 1.1}

Simplifying the expression, we get:

S50=1.15(1−1.150)−0.1S_{50} = \frac{1.1^{5}(1 - 1.1^{50})}{-0.1}

This is the approximate value of the given integral.

Comparing with the Given Options

Now, let's compare the approximate value of the integral with the given options:

A. ∫59.91.1x dx\int_5^{9.9} 1.1^x \, dx

B. ∫5101.1x dx\int_5^{10} 1.1^x \, dx

C. 110∫50991.1x dx\frac{1}{10} \int_{50}^{99} 1.1^x \, dx

The approximate value of the integral is closest to option C.

Conclusion

In this article, we have seen how to approximate a specific integral using a geometric series. We have also compared the approximate value of the integral with the given options and found that it is closest to option C. This demonstrates the power of geometric series in approximating integrals and highlights the importance of understanding these concepts in mathematics.

The Final Answer

The final answer is option C: 110∫50991.1x dx\frac{1}{10} \int_{50}^{99} 1.1^x \, dx.

Discussion

This problem is a great example of how geometric series can be used to approximate integrals. The key concept here is to recognize that the given integral can be rewritten as a geometric series and then use the formula for the sum of a geometric series to approximate the value of the integral.

Additional Resources

For more information on geometric series and their applications in mathematics, please refer to the following resources:

References

Introduction

In our previous article, we explored how to approximate a specific integral using a geometric series. In this article, we will answer some frequently asked questions about approximating integrals with geometric series.

Q: What is a geometric series?

A: A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: How do I recognize when an integral can be approximated with a geometric series?

A: To recognize when an integral can be approximated with a geometric series, look for the following characteristics:

  • The integral has a constant base and a variable exponent.
  • The variable exponent is a linear function of the variable of integration.
  • The integral has a finite number of terms.

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

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

Sn=a(1−rn)1−rS_n = \frac{a(1 - r^n)}{1 - r}

where aa is the first term, rr is the common ratio, and nn is the number of terms.

Q: How do I apply the formula for the sum of a geometric series to approximate an integral?

A: To apply the formula for the sum of a geometric series to approximate an integral, follow these steps:

  1. Identify the first term, common ratio, and number of terms in the geometric series.
  2. Plug these values into the formula for the sum of a geometric series.
  3. Simplify the expression to get the approximate value of the integral.

Q: What are some common applications of geometric series in mathematics?

A: Geometric series have many applications in mathematics, including:

  • Approximating integrals
  • Solving differential equations
  • Modeling population growth
  • Analyzing financial data

Q: What are some common mistakes to avoid when approximating integrals with geometric series?

A: Some common mistakes to avoid when approximating integrals with geometric series include:

  • Not recognizing when an integral can be approximated with a geometric series
  • Not identifying the first term, common ratio, and number of terms correctly
  • Not simplifying the expression correctly

Q: How do I know when to use a geometric series to approximate an integral versus another method?

A: To determine when to use a geometric series to approximate an integral versus another method, consider the following factors:

  • The complexity of the integral
  • The desired level of accuracy
  • The computational resources available

Conclusion

In this article, we have answered some frequently asked questions about approximating integrals with geometric series. We hope that this information has been helpful in understanding this important concept in mathematics.

Additional Resources

For more information on geometric series and their applications in mathematics, please refer to the following resources:

References