Calculating the Sum of an Arithmetic Progression Efficiently

Arithmetic progressions (APs) play a significant role in a wide array of mathematical applications. They are used in various fields such as finance, physics, and computer science. One such example is the series 21, 29, 37, 45, ..., 1061. In this article, we will explore how to calculate the sum of such a series using a straightforward approach.

Introduction to Arithmetic Progression

An arithmetic progression is a sequence in which the difference between the consecutive terms is constant. This constant difference is referred to as the common difference. For the given series 21, 29, 37, 45, ..., 1061, the first term a is 21, and the common difference d is 29 - 21 8.

Formulas for Arithmetic Progression

The formula for the nth term of an arithmetic progression is given by:

tn an-1d

The formula for the sum of the first n terms of an arithmetic progression is:

Sn n/2[2a (n - 1)d]

Calculation Steps

Given the series 21, 29, 37, 45, ..., 1061, we need to find the number of terms n, and then calculate the sum using the summation formula.

Step 1: Determine the Number of Terms n

The formula to find the number of terms n is:

tn an-1d

Here, tn is the last term, an-1 is the first term, and d is the common difference:

1061 21 (n - 1)8

1061 - 21 8(n - 1)

1040 8(n - 1)

1040 / 8 (n - 1)

130 (n - 1)

n 131

Step 2: Calculate the Sum of the Series

Using the sum formula:

Sn n/2[2a (n - 1)d]

Substitute the values:

S131 131/2[2 * 21 (131 - 1)8]

S131 131/2[42 1040]

S131 131/2[1082]

S131 131 * 541

S131 70871

Conclusion

By following a systematic approach, we can efficiently calculate the sum of the series 21, 29, 37, 45, ..., 1061. The sum of this series is 70871.

Understanding the formulas and methods for calculating sums of arithmetic progressions is essential for solving complex mathematical problems and extracting useful insights.