Sum of the First 10 Terms of an Arithmetic Sequence: A Comprehensive Guide
In mathematics, an arithmetic sequence, also known as an arithmetic progression (AP), is a sequence of numbers such that the difference between any two successive members is a constant. This constant difference is known as the common difference (d). Understanding how to calculate the sum of the first n terms in an arithmetic sequence is a fundamental skill in algebra and has numerous practical applications in fields such as finance, physics, and computer science.
Given Information
Consider an arithmetic sequence where the first term a1 is 22 and the 10th term a10 is 40. We aim to find the sum of the first 10 terms of this sequence.
Step-by-Step Calculation
1. **Identify the Common Difference (d):** The common difference can be found using the formula for the nth term of an arithmetic sequence: an a1 (n - 1)d. For the 10th term: a10 a1 9d. Substitute the known values: 40 22 9d. Solve for d: 40 - 22 9d → 18 9d → d 2.
Sum of the First 10 Terms
The sum of the first n terms of an arithmetic sequence is given by the formula: Sn n/2 [2a (n - 1)d] or Sn n/2 (a l), where a is the first term and l is the last term.
Using the First and Last Term Formula
Substitute n 10, a 22, and l 40 into the formula: S10 10/2 [22 40]. Simplify: S10 5 [62] 310.Using the Standard Sum Formula
Alternatively, use the standard sum formula: Sn n/2 (a l). Substitute the known values: S10 10/2 (22 40) 5 [62] 310.Verification
To ensure accuracy, we can verify the common difference and the sum using multiple steps:
The common difference d: 40 - 22 18, and 18/9 2. The 10th term: 22 (10 - 1)2 22 18 40. The sum verification: 22 24 26 28 30 32 34 36 38 40 310.Conclusion
The sum of the first 10 terms of the arithmetic sequence is 310. This method can be applied to any arithmetic sequence by substituting the appropriate values for the first term and the common difference.
Further Reading and Resources
To deepen your understanding of arithmetic sequences and related concepts:
Explore more on arithmetic sequences with interactive examples. Read about other series and sequences. Learn advanced applications of arithmetic sequences.