(iii) -5 + (-8) + (-11) ++ (-230) In An AP:​

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Introduction

Arithmetic Progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term. This constant is called the common difference. In this article, we will discuss how to find the sum of an arithmetic progression, with a focus on the given problem: -5 + (-8) + (-11) ++ (-230) In an AP.

Understanding Arithmetic Progression

An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term. For example, 2, 5, 8, 11, 14, ... is an arithmetic progression with a common difference of 3. The formula to find the nth term of an arithmetic progression is given by: an = a + (n - 1)d, where an is the nth term, a is the first term, n is the number of terms, and d is the common difference.

Finding the Sum of an Arithmetic Progression

The sum of an arithmetic progression can be found using the formula: Sn = n/2 (a + l), where Sn is the sum of the first n terms, a is the first term, l is the last term, and n is the number of terms. However, in this problem, we are given a sequence of numbers and asked to find the sum. We need to find the common difference and the number of terms in the sequence.

Finding the Common Difference

To find the common difference, we can subtract any two consecutive terms in the sequence. Let's subtract the second term from the first term: -8 - (-5) = -3. This means that the common difference is -3.

Finding the Number of Terms

To find the number of terms, we can use the formula: an = a + (n - 1)d. We know that the last term is -230, the first term is -5, and the common difference is -3. We can plug in these values into the formula and solve for n: -230 = -5 + (n - 1)(-3). Simplifying the equation, we get: -225 = -3n + 3. Subtracting 3 from both sides, we get: -228 = -3n. Dividing both sides by -3, we get: n = 76.

Finding the Sum

Now that we have found the common difference and the number of terms, we can find the sum of the sequence. We can use the formula: Sn = n/2 (a + l). Plugging in the values, we get: Sn = 76/2 (-5 + (-230)). Simplifying the equation, we get: Sn = 38 (-235). Multiplying the numbers, we get: Sn = -8950.

Conclusion

In this article, we discussed how to find the sum of an arithmetic progression, with a focus on the given problem: -5 + (-8) + (-11) ++ (-230) In an AP. We found the common difference to be -3 and the number of terms to be 76. We then used the formula to find the sum of the sequence, which is -8950.

Frequently Asked Questions

  • What is an arithmetic progression? An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term.
  • How do I find the sum of an arithmetic progression? You can use the formula: Sn = n/2 (a + l), where Sn is the sum of the first n terms, a is the first term, l is the last term, and n is the number of terms.
  • How do I find the common difference in an arithmetic progression? You can subtract any two consecutive terms in the sequence to find the common difference.
  • How do I find the number of terms in an arithmetic progression? You can use the formula: an = a + (n - 1)d, where an is the nth term, a is the first term, n is the number of terms, and d is the common difference.

Example Problems

  • Find the sum of the arithmetic progression: 2, 5, 8, 11, 14, ... First, we need to find the common difference, which is 3. Then, we need to find the number of terms, which is 5. Finally, we can use the formula to find the sum: Sn = 5/2 (2 + 14) = 60.
  • Find the sum of the arithmetic progression: -10, -7, -4, -1, 2, ... First, we need to find the common difference, which is 3. Then, we need to find the number of terms, which is 5. Finally, we can use the formula to find the sum: Sn = 5/2 (-10 + 2) = -24.

Summary

In this article, we discussed how to find the sum of an arithmetic progression, with a focus on the given problem: -5 + (-8) + (-11) ++ (-230) In an AP. We found the common difference to be -3 and the number of terms to be 76. We then used the formula to find the sum of the sequence, which is -8950. We also provided example problems and frequently asked questions to help readers understand the concept better.

Frequently Asked Questions

Q: What is an arithmetic progression?

A: An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term.

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

A: You can use the formula: Sn = n/2 (a + l), where Sn is the sum of the first n terms, a is the first term, l is the last term, and n is the number of terms.

Q: How do I find the common difference in an arithmetic progression?

A: You can subtract any two consecutive terms in the sequence to find the common difference.

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

A: You can use the formula: an = a + (n - 1)d, where an is the nth term, a is the first term, n is the number of terms, and d is the common difference.

Q: What is the formula to find the nth term of an arithmetic progression?

A: The formula to find the nth term of an arithmetic progression is: an = a + (n - 1)d, where an is the nth term, a is the first term, n is the number of terms, and d is the common difference.

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

A: You can use the formula: Sn = n/2 (a + l), where Sn is the sum of the first n terms, a is the first term, l is the last term, and n is the number of terms.

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

A: You can use the formula: Sn = a / (1 - r), where Sn is the sum of the first n terms, a is the first term, and r is the common ratio.

Q: What is the formula to find the sum of an arithmetic progression with a common difference of 0?

A: The formula to find the sum of an arithmetic progression with a common difference of 0 is: Sn = n/2 (a + a) = n/2 (2a) = na.

Q: How do I find the sum of an arithmetic progression with a common difference of 1?

A: You can use the formula: Sn = n/2 (a + l), where Sn is the sum of the first n terms, a is the first term, l is the last term, and n is the number of terms.

Q: How do I find the sum of an arithmetic progression with a common difference of -1?

A: You can use the formula: Sn = n/2 (a + l), where Sn is the sum of the first n terms, a is the first term, l is the last term, and n is the number of terms.

Q: What is the formula to find the sum of an arithmetic progression with a common difference of 2?

A: The formula to find the sum of an arithmetic progression with a common difference of 2 is: Sn = n/2 (a + l), where Sn is the sum of the first n terms, a is the first term, l is the last term, and n is the number of terms.

Q: How do I find the sum of an arithmetic progression with a common difference of -2?

A: You can use the formula: Sn = n/2 (a + l), where Sn is the sum of the first n terms, a is the first term, l is the last term, and n is the number of terms.

Example Problems

Q: Find the sum of the arithmetic progression: 2, 5, 8, 11, 14, ...

A: First, we need to find the common difference, which is 3. Then, we need to find the number of terms, which is 5. Finally, we can use the formula to find the sum: Sn = 5/2 (2 + 14) = 60.

Q: Find the sum of the arithmetic progression: -10, -7, -4, -1, 2, ...

A: First, we need to find the common difference, which is 3. Then, we need to find the number of terms, which is 5. Finally, we can use the formula to find the sum: Sn = 5/2 (-10 + 2) = -24.

Q: Find the sum of the arithmetic progression: 1, 4, 7, 10, 13, ...

A: First, we need to find the common difference, which is 3. Then, we need to find the number of terms, which is 5. Finally, we can use the formula to find the sum: Sn = 5/2 (1 + 13) = 42.

Tips and Tricks

Tip 1: Make sure to find the common difference before finding the sum.

Finding the common difference is crucial in finding the sum of an arithmetic progression.

Tip 2: Use the formula to find the sum of an arithmetic progression.

The formula to find the sum of an arithmetic progression is: Sn = n/2 (a + l), where Sn is the sum of the first n terms, a is the first term, l is the last term, and n is the number of terms.

Tip 3: Check your work by plugging in the values into the formula.

Make sure to check your work by plugging in the values into the formula to ensure that you get the correct answer.

Conclusion

In this article, we discussed how to find the sum of an arithmetic progression, with a focus on the given problem: -5 + (-8) + (-11) ++ (-230) In an AP. We also provided example problems and frequently asked questions to help readers understand the concept better. We hope that this article has been helpful in understanding how to find the sum of an arithmetic progression.