Understanding the Sum of the First N Positive Integers using Sigma Notation

The expression (Sigma_{i1}^{N} i) represents the sum of the first N positive integers. This concept is fundamental in mathematics and has wide applications in various fields. This article will delve into the details of how to compute this sum and provide a proof of the formula using mathematical induction.

Basic Explanation

The mathematical symbol “Σ” is used to denote a sum. This means we only need to perform addition operations during the calculation. The subscript “i1” indicates the starting value of the integer sequence, and the superscript “N” represents the last value in the sequence.

For example, to find the sum of the first 5 positive integers, we have:

(Sigma_{i1}^{5} i 1 2 3 4 5 15)

Using the derived formula, (frac{N(N 1)}{2}), we can calculate the same sum:

(frac{5(5 1)}{2} frac{5 times 6}{2} 15)

Note: For any positive integer N, you can use this formula to find the sum of the first N integers.

Step-by-Step Calculation

Let's denote the sum of the first N positive integers by the symbol S. To compute S, we can write it as follows:

(S 1 2 3 ... (N-1) N)

When we add the sequence in reverse order, we get:

(S N (N-1) (N-2) ... 3 2 1)

Adding these two equations, we get:

(2S (1 N) (2 (N-1)) (3 (N-2)) ... (N-1 2) (N 1))

Each pair sums to N 1, and there are N such pairs:

(2S N times (N 1))

Solving for S, we get:

(S frac{N(N 1)}{2})

Mathematical Induction Proof

To prove that the formula (S_N sum_{i1}^N i frac{N(N 1)}{2}) holds for all integers N geq 1, we will use mathematical induction.

Base Case

For N 1:

(S_1 1 frac{1(1 1)}{2} 1)

The base case is true.

Induction Hypothesis

Assume that the formula holds for N. That is:

(S_N 1 2 3 ... N frac{N(N 1)}{2})

Inductive Step

We need to show that the formula holds for N 1. That is:

(S_{N 1} 1 2 3 ... N (N 1) frac{(N 1)(N 2)}{2})

Using the induction hypothesis:

(S_{N 1} S_N (N 1) frac{N(N 1)}{2} (N 1))

We can simplify the right-hand side:

(frac{N(N 1) 2(N 1)}{2} frac{(N 1)(N 2)}{2})

Thus, the formula holds for N 1.

By the principle of mathematical induction, the formula (S_N sum_{i1}^N i frac{N(N 1)}{2}) is true for all integers N geq 1.

Conclusion

In summary, the sum of the first N positive integers is calculated using the formula (frac{N(N 1)}{2}). This formula can be derived both conceptually and through mathematical induction, making it a robust and reliable method for solving a wide range of problems in mathematics and beyond.