Show that the formula 2+4+6+⋯+2 n=n^2+n+2 obeys Condition II of the Principle of Mathematical In

step 1 The given statement is 2 plus 4 plus 6 plus up to 2n is equal to n squared plus n plus 2. Let's

Show that the formula 2+4+6+⋯+2 n=n^2+n+2 obeys Condition II...

Show that the formula
$$
2+4+6+\cdots+2 n=n^2+n+2
$$
obeys Condition II of the Principle of Mathematical Induction. That is, show that if the formula is true for some $k$, it is also true for $k+1$. Then show that the formula is false for $n=1$ (or for any other choice of $n$ ).

Key Concepts

Counterexample

A counterexample is an example that disproves a general statement by showing that it fails under specific conditions. In the context of proofs by induction or any general claim, presenting a counterexample, such as a case where the base step does not hold, demonstrates that the proposed formula or claim is incorrect.

Base Case Verification

Verifying the base case involves checking the validity of the statement for the initial value, typically n=1. This step is essential in any induction proof because if the statement fails at the base level, then the inductive process cannot begin, rendering the overall claim invalid.

Inductive Step

The inductive step is the part of the proof where one assumes the statement is true for some arbitrary natural number k (this is the inductive hypothesis) and then uses that assumption to show that the statement must also hold for k+1. This step is crucial because it establishes the link between successive cases and is essential for extending the truth of the statement to all natural numbers.

Principle of Mathematical Induction

Mathematical induction is a proof technique used to establish that a given statement holds for all natural numbers. It involves proving a base case to show the statement is true for an initial value and then performing an inductive step, where one assumes the statement is true for an arbitrary natural number k and proves it is true for k+1. This method is widely used to prove propositions involving sequences, series, or recursively defined structures.

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