Chapter 12.4, Problem 1AYU

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.

$2\text{ + 4 + 6 + }\mathrm{...}\text{ + 2}n\text{ = }n\left(n\text{ + 1}\right)$

To determine

To prove: The given statement $2 + 4 + 6 + \cdot \cdot \cdot + 2n = n(n + 1)$ is true for all natural numbers $n$ using the Principle of Mathematical Induction.

Answer to Problem 1AYU

As the statement is true for the natural number $(k + 1)$ terms, hence the statement is true for all natural numbers.

Explanation of Solution

Given:

Statements says the series $2 + 4 + 6 + \cdot \cdot \cdot + 2n = n(n + 1)$ is true for all natural number.

Formula used:

The Principle of Mathematical Induction

Suppose that the following two conditions are satisfied with regard to a statement about natural numbers:

CONDITION I: The statement is true for the natural number 1.

CONDITION II: If the statement is true for some natural number $k$, it is also true for the next natural number $k + 1.$ Then the statement is true for all natural numbers.

Proof:

Consider the statement $2 + 4 + 6 + \cdot \cdot \cdot + 2n = n(n + 1)$   -----(1)

Step 1: Show that statement (1) is true for $n = 1$.

That is $2(1) = 1(1 + 1).$ Hence the statement is true for natural number $n = 1$.

Step 2: Assume that the statement is true for some natural number $k$.

That is $2 + 4 + 6 + \cdot \cdot \cdot + 2k = k(k + 1)$   -----(1)

Step 3: Prove that the statement is true for the next natural number $k + 1$.

That is, to prove that $2 + 4 + 6 + \cdot \cdot \cdot + 2(k + 1) = (k + 1)(k + 2)$

Consider $2 + 4 + 6 + \cdot \cdot \cdot + 2(k + 1)$

=$2 + 4 + 6 + \cdot \cdot \cdot + 2k + 2(k + 1)$

$k(k + 1) + 2(k + 1)$ [Substituting equation (1)]

$= (k + 1)(k + 2)$

As the statement is true for the natural number $(k + 1)$ terms, hence the statement is true for all natural numbers.