Calculating the Sum of the First 20 Terms of an Arithmetic Sequence

Introduction

An arithmetic sequence is a sequence of numbers where the difference between any two successive members is constant. The sum of the first n terms of an arithmetic sequence is a common topic in mathematics, especially in calculus and algebra. This article will guide you through calculating the sum of the first 20 terms of an arithmetic sequence, given that the fourth term is 6 and the eleventh term is 30.

Understanding the Problem

We are given that T4 6 and T11 30. Let’s denote the first term by A and the common difference by d. Using the general formula of an arithmetic sequence, Tn A (n-1)d, we can set up the following equations:

T4 A 3d 6 … (1)

T11 A 10d 30 … (2)

Solving for d and A

By subtracting equation (1) from equation (2), we can eliminate A and solve for d

A 10d - (A 3d) 30 - 6

7d 24

d 24/7

Substitute this value of d back into equation (1):

A 3(24/7) 6

A 72/7 6

A 6 - 72/7 (42 - 72)/7 -30/7

Calculating the Sum of the First 20 Terms

The sum of the first n terms of an arithmetic sequence is given by the formula:

Sn n/2 * [2A (n-1)d]

For our problem, we need to find the sum of the first 20 terms (n 20):

S20 20/2 * [2(-30/7) 19(24/7)]

S20 10 * [-60/7 456/7]

S20 10 * (456/7 - 60/7) 10 * (396/7)

S20 3960/7

The sum of the first 20 terms of the sequence is 3960/7.

Conclusion

In summary, by using the properties and formulas of an arithmetic sequence, we were able to determine that the common difference is 24/7 and the first term is -30/7. Utilizing these values, we derived the sum of the first 20 terms to be 3960/7. This calculation exemplifies the application of basic arithmetic sequence concepts and can be expanded to more complex problems involving sequences and series.