The Arithmetic Sequence Formula: Understanding the Sum of Terms

In mathematics, an arithmetic sequence is a series of numbers where the difference between any two consecutive terms is constant. This constant difference is known as the common difference. For example, the sequence 2, 5, 8, 11, 14… is an arithmetic sequence with a common difference of 3. Understanding arithmetic sequences is crucial in various fields, including finance, physics, and computer science.

What is the Formula for the Sum of an Arithmetic Sequence?

The sum of an arithmetic sequence, also known as an arithmetic series, can be calculated using a specific formula. This formula provides a concise and efficient way to determine the total sum of a sequence without having to add up all the individual terms. The formula is:

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

Where:

• Sn represents the sum of the first n terms of the arithmetic sequence.
• n is the number of terms in the sequence.
• a is the first term of the sequence.
• l is the last term of the sequence.

Understanding the Formula

The formula essentially takes the average of the first and last terms and multiplies it by the number of terms. This approach makes sense because the terms in an arithmetic sequence increase or decrease at a constant rate. Therefore, the average of the first and last terms represents the average value of all the terms in the sequence.

Example: Finding the Sum of an Arithmetic Sequence

Let’s say we have the arithmetic sequence 3, 7, 11, 15, 19. We want to find the sum of these five terms.

Here’s how we can apply the formula:

• n = 5 (number of terms)
• a = 3 (first term)
• l = 19 (last term)

Plugging these values into the formula, we get:

Sn = (5/2) * (3 + 19)

Sn = 2.5 * 22

Sn = 55

Therefore, the sum of the arithmetic sequence 3, 7, 11, 15, 19 is 55.

Practice Questions

To solidify your understanding, try solving these practice questions:

1. Find the sum of the first 10 terms of the arithmetic sequence 1, 4, 7, 10…
2. What is the sum of the arithmetic sequence 2, 6, 10, 14… up to the 15th term?
3. An arithmetic sequence has a first term of 5 and a common difference of 2. Calculate the sum of the first 20 terms.

Conclusion

The arithmetic sequence formula provides a powerful tool for calculating the sum of any arithmetic sequence. By understanding the formula and its derivation, you can efficiently determine the total sum of a sequence, regardless of its length. This knowledge is invaluable in various mathematical applications, making arithmetic sequences a fundamental concept in the study of mathematics.