Evaluate The Following Expression Without Using A Calculator:$ (553 + 554 + 555 + \ldots + 600) - (201 + 202 + 203 + \ldots + 248) $

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Introduction

In this article, we will evaluate the given mathematical expression without using a calculator. The expression involves the sum of consecutive numbers, and we will use algebraic techniques to simplify it.

The Expression

The given expression is:

$ (553 + 554 + 555 + \ldots + 600) - (201 + 202 + 203 + \ldots + 248) $

This expression involves the sum of two arithmetic series. The first series starts from 553 and ends at 600, while the second series starts from 201 and ends at 248.

Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. The sum of an arithmetic series can be calculated using the formula:

$ S_n = \frac{n}{2} (a_1 + a_n) $

where $ S_n $ is the sum of the series, $ n $ is the number of terms, $ a_1 $ is the first term, and $ a_n $ is the last term.

Evaluating the First Series

The first series starts from 553 and ends at 600. To evaluate this series, we need to find the number of terms, the first term, and the last term.

The number of terms in the series can be calculated as:

$ n = a_n - a_1 + 1 $

where $ a_n $ is the last term and $ a_1 $ is the first term.

In this case, $ a_n = 600 $ and $ a_1 = 553 $, so:

$ n = 600 - 553 + 1 = 48 $

Now, we can use the formula for the sum of an arithmetic series:

$ S_1 = \frac{48}{2} (553 + 600) $

$ S_1 = 24 (1153) $

$ S_1 = 27672 $

Evaluating the Second Series

The second series starts from 201 and ends at 248. To evaluate this series, we need to find the number of terms, the first term, and the last term.

The number of terms in the series can be calculated as:

$ n = a_n - a_1 + 1 $

where $ a_n $ is the last term and $ a_1 $ is the first term.

In this case, $ a_n = 248 $ and $ a_1 = 201 $, so:

$ n = 248 - 201 + 1 = 48 $

Now, we can use the formula for the sum of an arithmetic series:

$ S_2 = \frac{48}{2} (201 + 248) $

$ S_2 = 24 (449) $

$ S_2 = 10776 $

Evaluating the Expression

Now that we have evaluated both series, we can substitute the values into the original expression:

$ (553 + 554 + 555 + \ldots + 600) - (201 + 202 + 203 + \ldots + 248) $

$ = 27672 - 10776 $

$ = 16900 $

Conclusion

In this article, we evaluated the given mathematical expression without using a calculator. We used algebraic techniques to simplify the expression and found that the value of the expression is 16900.

Frequently Asked Questions

Q: What is the formula for the sum of an arithmetic series?

A: The formula for the sum of an arithmetic series is:

$ S_n = \frac{n}{2} (a_1 + a_n) $

where $ S_n $ is the sum of the series, $ n $ is the number of terms, $ a_1 $ is the first term, and $ a_n $ is the last term.

Q: How do I find the number of terms in an arithmetic series?

A: The number of terms in an arithmetic series can be calculated as:

$ n = a_n - a_1 + 1 $

where $ a_n $ is the last term and $ a_1 $ is the first term.

Q: What is the difference between an arithmetic series and a geometric series?

A: An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. A geometric series is a sequence of numbers in which the ratio between any two consecutive terms is constant.

References

  • [1] "Arithmetic Series" by Math Open Reference. Retrieved February 25, 2024.
  • [2] "Geometric Series" by Math Open Reference. Retrieved February 25, 2024.

Keywords

  • arithmetic series
  • geometric series
  • sum of an arithmetic series
  • number of terms in an arithmetic series
  • difference between an arithmetic series and a geometric series

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Introduction

In our previous article, we evaluated the given mathematical expression without using a calculator. We used algebraic techniques to simplify the expression and found that the value of the expression is 16900. In this article, we will answer some frequently asked questions about arithmetic series.

Q&A

Q: What is an arithmetic series?

A: An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant.

Q: How do I find the sum of an arithmetic series?

A: The sum of an arithmetic series can be calculated using the formula:

$ S_n = \frac{n}{2} (a_1 + a_n) $

where $ S_n $ is the sum of the series, $ n $ is the number of terms, $ a_1 $ is the first term, and $ a_n $ is the last term.

Q: How do I find the number of terms in an arithmetic series?

A: The number of terms in an arithmetic series can be calculated as:

$ n = a_n - a_1 + 1 $

where $ a_n $ is the last term and $ a_1 $ is the first term.

Q: What is the difference between an arithmetic series and a geometric series?

A: An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. A geometric series is a sequence of numbers in which the ratio between any two consecutive terms is constant.

Q: Can I use the formula for the sum of an arithmetic series if the series is not in order?

A: No, the formula for the sum of an arithmetic series only works if the series is in order. If the series is not in order, you will need to rearrange the terms before using the formula.

Q: How do I find the average of an arithmetic series?

A: The average of an arithmetic series can be calculated by dividing the sum of the series by the number of terms:

$ \text{Average} = \frac{S_n}{n} $

Q: Can I use the formula for the sum of an arithmetic series if the series has a negative difference?

A: Yes, the formula for the sum of an arithmetic series works even if the series has a negative difference.

Q: How do I find the sum of a finite arithmetic series?

A: The sum of a finite arithmetic series can be calculated using the formula:

$ S_n = \frac{n}{2} (a_1 + a_n) $

where $ S_n $ is the sum of the series, $ n $ is the number of terms, $ a_1 $ is the first term, and $ a_n $ is the last term.

Q: How do I find the sum of an infinite arithmetic series?

A: The sum of an infinite arithmetic series can be calculated using the formula:

$ S = \frac{a_1}{1 - r} $

where $ S $ is the sum of the series, $ a_1 $ is the first term, and $ r $ is the common difference.

Conclusion

In this article, we answered some frequently asked questions about arithmetic series. We covered topics such as the sum of an arithmetic series, the number of terms in an arithmetic series, and the difference between an arithmetic series and a geometric series.

Frequently Asked Questions

Q: What is the formula for the sum of an arithmetic series?

A: The formula for the sum of an arithmetic series is:

$ S_n = \frac{n}{2} (a_1 + a_n) $

where $ S_n $ is the sum of the series, $ n $ is the number of terms, $ a_1 $ is the first term, and $ a_n $ is the last term.

Q: How do I find the number of terms in an arithmetic series?

A: The number of terms in an arithmetic series can be calculated as:

$ n = a_n - a_1 + 1 $

where $ a_n $ is the last term and $ a_1 $ is the first term.

Q: What is the difference between an arithmetic series and a geometric series?

A: An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. A geometric series is a sequence of numbers in which the ratio between any two consecutive terms is constant.

References

  • [1] "Arithmetic Series" by Math Open Reference. Retrieved February 25, 2024.
  • [2] "Geometric Series" by Math Open Reference. Retrieved February 25, 2024.

Keywords

  • arithmetic series
  • geometric series
  • sum of an arithmetic series
  • number of terms in an arithmetic series
  • difference between an arithmetic series and a geometric series