Question

Describe induction.

Short answer

Induction is a technique used to prove statements for all natural numbers, comprising of the base case and inductive step. Here, we verify the formula for the sum of the first 'n' natural numbers, \(S = \frac{n(n+1)}{2}\), by establishing the base case for n=1 and assuming the statement's validity for an arbitrary natural number 'k.' We then prove the statement for 'k+1' and, by the principle of mathematical induction, confirm the formula's accuracy for all natural numbers greater than or equal to the base case (n=1).

Step-by-step solution

Understand the Problem Statement

We will use mathematical induction to verify the formula for the sum of the first 'n' natural numbers: \(S = \frac{n(n+1)}{2}\)

Establish the Base Case

When the problem involves natural numbers, the base case is typically 1 or 0. In this example, we will start with 1. \(S = 1\) Using the formula: \(S = \frac{n(n+1)}{2}\) Plug in n=1: \(S = \frac{1(1+1)}{2}\) Calculate the sum: \(S=1\) Since the formula yields the correct sum for n=1, the base case is established.

Assume the Inductive Hypothesis

For the inductive step, assume that the formula holds true for some arbitrary natural number 'k': \(S_k = \frac{k(k+1)}{2}\)

Prove the Statement for 'k+1'

We want to show that the formula also holds true for 'k+1': \(S_{k+1} = \frac{(k+1)(k+1+1)}{2}\) Start by expressing \(S_{k+1}\) in terms of \(S_k\): \(S_{k+1} = S_k + (k+1)\) Now, substitute the inductive hypothesis into the equation: \(\frac{(k+1)(k+1+1)}{2} = \frac{k(k+1)}{2} + (k+1)\)

Simplify and Verify the Inductive Step

Simplify the equation and verify that both sides are equal: \(\frac{(k+1)(k+2)}{2} = \frac{k(k+1) + 2(k+1)}{2}\) \(\frac{(k+1)(k+2)}{2} = \frac{(k+1)(k+2)}{2}\) Since both sides are equal, the inductive step is proven.

Conclude the Proof

We have demonstrated that the formula for the sum of the first 'n' natural numbers holds true for the base case and the inductive step. By the principle of mathematical induction, the formula holds true for all natural numbers greater than or equal to the base case (n=1): \(S = \frac{n(n+1)}{2}\)