Use the Principle of Mathematical Induction to show that the given statement is true for all nat

step 1 In this problem we have to prove the given statement by using principle of mathematical inductio

Use the Principle of Mathematical Induction to show that the...

Key Concepts

Principle of Mathematical Induction

This is a proof technique used to establish that a property holds for all natural numbers. It involves two main steps: first, showing the property holds for an initial value (usually n = 1), and second, proving that if the property holds for an arbitrary natural number k, it then holds for k + 1. This method provides a systematic way to extend the validity of a statement to all elements in an infinite set.

Base Case

The base case is the first step in an induction proof where the statement is verified for the initial natural number, often n = 1. Establishing the base case ensures that the sequence of logic used in the induction process has a valid starting point.

Inductive Hypothesis

In the induction proof, the inductive hypothesis is the assumption that the given statement is true for some arbitrary natural number k. This assumption is used as a bridge to prove that the statement must also be true for the next number, k + 1.

Inductive Step

This is the part of the proof where one shows that if the statement holds for an arbitrary natural number k (as assumed in the inductive hypothesis), then it also holds for k + 1. Successful completion of this step allows one to use the principle of induction to conclude that the statement is true for all natural numbers.

Divisibility

Divisibility is a mathematical concept that relates to whether one integer can be divided by another without a remainder. In this context, demonstrating that an algebraic expression is divisible by a number, such as 2, entails showing that the expression produces an integer multiple of that number, confirming the evenness of the expression.

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