The Summation of Sequences in Mathematics

In the field of mathematics, summation is a fundamental concept that refers to the process of adding a series of numbers or terms. This article will delve into different methods to calculate the sum of specific sequences, particularly focusing on the sum of the alternating series from 12 to 100. We will explore arithmetic progressions and other useful techniques to simplify such calculations.

Classic Example: The Summation of Integers from 1 to 100

The Gaussian Sum is a classic problem often attributed to the famous German mathematician Carl Friedrich Gauss. While he was still a child, his teacher assigned him the task of adding all integers from 1 to 100 as a means of keeping the class busy. Instead of the long, boring task, Gauss devised a clever method to find the sum quickly.

To solve this, Gauss paired the numbers from the beginning and the end of the sequence:

Such pairing occurs 50 times, and thus the sum is calculated as follows:

101 times; 50 5050

Generalizing the Approach

More generally, the sum of integers from 1 to n is given by the formula:

S nn 1/2

This formula provides a quick way to determine the sum for any given range of integers.

Altelling Summation: 12 - 34 - 56 - 78 ... 100

Step 1: Sum of Even Numbers from 2 to 100

The even numbers from 2 to 100 form an arithmetic progression (AP). The sum of an arithmetic progression can be calculated by the formula:

Sum of n terms n/2 (a L)

Here, n 50 (since there are 50 even numbers), a 2 (the first term), and L 100 (the last term).

Sum of 50 terms 50/2 (2 100) 25 times; 102 2550

Step 2: Sum of Remaining Terms

The remaining terms are the odd numbers from 3 to 99, also forming an arithmetic progression. The sum of 49 terms (since the first term 1 is excluded) can be calculated as follows:

Sum of 49 terms 49/2 times; (3 99) 49/2 times; 102 49 times; 51 2499

Step 3: Final Calculation

Adding the result of the even numbers' sum and the odd numbers' sum:

2550 - 2499 1 (the very first term) 2550 - 2499 1 52

Therefore, 12 - 34 - 56 - 78 ... 100 52

Conclusion

Through the use of arithmetic progressions and clever pairing techniques, we can efficiently compute the sum of sequences that may seem daunting at first glance. These methods not only simplify complex calculations but also illustrate the elegance and power of mathematical reasoning.