Formula for the Sum of Integers in Arithmetic Progression

Oct 13, 2025 by miodeya.com 137 views

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Introduction

The sum of integers in an arithmetic progression is a fundamental concept in mathematics, with numerous applications in various fields, including algebra, geometry, and analysis. In this article, we will delve into the world of arithmetic progressions and explore the sum of integers in such sequences. We will also provide examples and formulas to help readers understand and calculate the sum of integers in arithmetic progressions.

What is an Arithmetic Progression?

An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. For example, the sequence 2, 5, 8, 11, 14, ... is an arithmetic progression with a common difference of 3. The sum of integers in an arithmetic progression can be calculated using the formula:

Sum = (n/2) * (a + l)

where n is the number of terms, a is the first term, and l is the last term.

Formula for the Sum of Integers in Arithmetic Progression

The formula for the sum of integers in an arithmetic progression is given by:

Sum = (n/2) * (a + l)

where n is the number of terms, a is the first term, and l is the last term. This formula can be derived by adding the terms of the arithmetic progression and using the formula for the sum of an arithmetic series.

Example 1: Sum of Integers from 26 to 63

The sum of integers from 26 to 63 is given by:

Sum = (63 - 26 + 1)/2 * (26 + 63)

Sum = (38/2) * (89)

Sum = 19 * 89

Sum = 1691

Example 2: Sum of Integers from 58 to 71

The sum of integers from 58 to 71 is given by:

Sum = (71 - 58 + 1)/2 * (58 + 71)

Sum = (14/2) * (129)

Sum = 7 * 129

Sum = 903

Example 3: Sum of Integers from 1 to 100

The sum of integers from 1 to 100 is given by:

Sum = (100 - 1 + 1)/2 * (1 + 100)

Sum = (100/2) * (101)

Sum = 50 * 101

Sum = 5050

Example 4: Sum of Integers from 25 to 50

The sum of integers from 25 to 50 is given by:

Sum = (50 - 25 + 1)/2 * (25 + 50)

Sum = (26/2) * (75)

Sum = 13 * 75

Sum = 975

Conclusion

In conclusion, the sum of integers in an arithmetic progression can be calculated using the formula (n/2) * (a + l), where n is the number of terms, a is the first term, and l is the last term. We have provided examples and formulas to help readers understand and calculate the sum of integers in arithmetic progressions. The sum of integers in an arithmetic progression has numerous applications in various fields, including algebra, geometry, and analysis.

References

[1] "Arithmetic Progression" by Math Open Reference
[2] "Sum of an Arithmetic Series" by Math Is Fun
[3] "Arithmetic Progression Formula" by BYJU'S

Introduction

In our previous article, we explored the concept of arithmetic progressions and the formula for the sum of integers in such sequences. In this article, we will answer some frequently asked questions about the sum of integers in arithmetic progressions.

Q: What is an arithmetic progression?

A: An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. For example, the sequence 2, 5, 8, 11, 14, ... is an arithmetic progression with a common difference of 3.

Q: How do I calculate the sum of integers in an arithmetic progression?

A: The formula for the sum of integers in an arithmetic progression is given by:

Sum = (n/2) * (a + l)

where n is the number of terms, a is the first term, and l is the last term.

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

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

Sum = (n/2) * (a + l)

where n is the number of terms, a is the first term, and l is the last term.

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

A: To find the number of terms in an arithmetic progression, you can use the formula:

n = (l - a)/d + 1

where l is the last term, a is the first term, and d is the common difference.

Q: What is the formula for the sum of integers from 1 to n?

A: The formula for the sum of integers from 1 to n is given by:

Sum = (n * (n + 1))/2

Q: How do I calculate the sum of integers from 25 to 50?

A: To calculate the sum of integers from 25 to 50, you can use the formula:

Sum = (50 - 25 + 1)/2 * (25 + 50)

Sum = (26/2) * (75)

Sum = 13 * 75

Sum = 975

Q: What is the sum of integers from 1 to 100?

A: The sum of integers from 1 to 100 is given by:

Sum = (100 - 1 + 1)/2 * (1 + 100)

Sum = (100/2) * (101)

Sum = 50 * 101

Sum = 5050

Q: How do I calculate the sum of integers in an arithmetic progression with a common difference of 2?

A: To calculate the sum of integers in an arithmetic progression with a common difference of 2, you can use the formula:

Sum = (n/2) * (a + l)

where n is the number of terms, a is the first term, and l is the last term.

Q: What is the sum of integers from 25 to 75?

A: The sum of integers from 25 to 75 is given by:

Sum = (75 - 25 + 1)/2 * (25 + 75)

Sum = (51/2) * (100)

Sum = 25.5 * 100

Sum = 2550

Conclusion

In conclusion, the sum of integers in an arithmetic progression can be calculated using the formula (n/2) * (a + l), where n is the number of terms, a is the first term, and l is the last term. We have answered some frequently asked questions about the sum of integers in arithmetic progressions and provided examples and formulas to help readers understand and calculate the sum of integers in arithmetic progressions.

References

[1] "Arithmetic Progression" by Math Open Reference
[2] "Sum of an Arithmetic Series" by Math Is Fun
[3] "Arithmetic Progression Formula" by BYJU'S