Find 1+2+3+4+...+ 100, the sum of the first 100 natural numbers.

Key Concepts

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence, where each term increases by a constant difference. In this case, the sequence is the first 100 natural numbers, which can be expressed as 1, 2, 3, ..., 100. The sum of an arithmetic series can be calculated using the formula S_n = n/2 * (a + l), where n is the number of terms, a is the first term, and l is the last term.

Formula for the Sum of Natural Numbers

The sum of the first n natural numbers can be calculated using the formula S_n = n(n + 1)/2. This formula provides a quick way to find the sum without having to add each number individually. For n = 100, substituting into the formula gives S_100 = 100(100 + 1)/2 = 5050, which is the desired sum.

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite number of statements. It involves two steps: proving the base case (usually for n=1) and then showing that if the statement holds for n=k, it also holds for n=k+1. This method can be used to verify the formula for the sum of the first n natural numbers, ensuring its validity for all natural numbers.