Question

Prove the following statements with either induction, strong induction or proof by smallest counterexample.Prove that \(1+2+3+4+\cdots+n=\frac{n^{2}+n}{2}\) for every positive integer \(n\).

Short answer

The formula \(1+2+3+4+...+n=n*(n+1)/2\) was proven to be correct for all positive integers n through induction.

Step-by-step solution

Base Case

First, it's necessary to show that the formula holds for the lowest possible value of n, which is 1 in this case. Substituting n = 1, \((1^2 + 1)/2 = 1\) which checks out because the left side for n=1 is also 1.

Inductive Hypothesis

Assume that the formula holds true for some positive integer n. So, \(1 + 2 + 3 + ... + n = n*(n+1)/2\).

Inductive Step

It's necessary to show that if the formula holds for n, it also holds for n + 1. So, we need to prove that \(1+2+3+...+n+(n+1) = (n+1)*((n+1)+1)/2 = (n+1)*(n+2)/2\). Substituting the inductive hypothesis into the equation, we get \(n*(n+1)/2 + n + 1 = (n+1)*(n+2)/2\). Simplifying both sides of the equation, they end up being the same, thereby proving the formula.

Key concepts

Proof Techniques

When it comes to verifying mathematical statements, specifically the validity of formulas and theorems, proof techniques are the tools mathematicians use to make logical arguments. One of the foundational techniques is mathematical induction, a powerful method most often used to prove statements involving an infinitely countable set like positive integers.

Understanding Mathematical Induction

Mathematical induction consists of two crucial steps — the base case and the inductive step. Initially, we establish the truth for a starting value, usually the smallest value for which the statement is supposed to hold. This is called the base case. The inductive step involves demonstrating that if the statement holds for an arbitrary instance, it must hold for the next instance.

The inductive step typically uses the assumption called the inductive hypothesis to show the truth of the statement for the next value, linking one case to the subsequent one, and thus laying the groundwork for a sort of domino effect. If both steps are confirmed, the result is a proof that covers all cases, effectively using a finite argument to prove an infinite number of instances.

Inductive Reasoning

While proof techniques are the skeleton of mathematical validation, inductive reasoning is the muscle that helps to flesh out proofs for numerous propositions. Inductive reasoning, as applied in mathematical proofs, is a process of making a generalization based on observing particular instances, or a way of thinking that suggests what's true for one case will be true for all.

Inductive Hypothesis Impact

In the context of the exercise, the inductive hypothesis — our assumption that the formula holds true for some positive integer — is essential. This hypothesis acts as a stepping stone; once we've established the base case, we propose that the formula is valid for a specific integer 'n' and then must show it's true for 'n+1'. The leap from 'n' to 'n+1' may seem small but represents the infinite continuum of natural numbers. By validating the hypothesis, we validate the pattern and therefore the general statement.

Sum of An Arithmetic Series

The provided exercise dives into a fundamental concept found in arithmetic and algebra: the sum of an arithmetic series. An arithmetic series is created by adding the terms of an arithmetic sequence, where each term is an equal distance from the next.

Formula for the Sum of an Arithmetic Series

The formula to find the sum of the first 'n' numbers, as presented in the exercise — \(\frac{n^2 + n}{2}\) — demonstrates a key characteristic of arithmetic progressions. The initial term here is 1, and the common difference between successive terms is also 1. The formula is both succinct and elegant, representing the sum of an arithmetic series in terms of 'n' itself. Once established, this formula can be used to calculate the sum of any series where each term increases by 1 from the previous term, making it an indispensable tool for solving myriad problems in both mathematics and applied disciplines.