Chapter 12.4, Problem 17AYU

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

$1\cdot \text{ 2 }+2\cdot 3+\text{ 3 }\cdot 4+\mathrm{...}+n\left(n+1\right)\text{ = }\frac{1}{3}n\left(n+1\right)\left(n\text{ + 2}\right)$

To determine

To prove: The given statement $1.2 + 2.3 + 3.4 + \cdots + n\left(n + 1\right) = \frac{1}{3} n\left(n + 1\right)\left(n + 2\right)$ is true for all natural numbers $n$ using the Principle of Mathematical Induction.

Answer to Problem 17AYU

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

Explanation of Solution

Given:

Statements says the series $1.2 + 2.3 + 3.4 + \cdots + n\left(n + 1\right) = \frac{1}{3} n\left(n + 1\right)\left(n + 2\right)$ 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 $1.2 + 2.3 + 3.4 + \cdots + n\left(n + 1\right) = \frac{1}{3} n\left(n + 1\right)\left(n + 2\right)$   -----(1)

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

That is $1\left(1 + 1\right) = \frac{1}{3}. \left(1 + 1\right)\left(1 + 2\right).$ 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 $1.2 + 2.3 + 3.4 + \cdots + k\left(k + 1\right) = \frac{1}{3} k\left(k + 1\right)\left(k + 2\right)$   -----(1)

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

That is, to prove that $1.2 + 2.3 + 3.4 + \cdots + \left(k + 1\right) + \left(k + 2\right) = \frac{1}{3}\left(k + 1\right)\left(k + 2\right)\left(k + 3\right)$

$1.2 + 2.3 + 3.4 + \cdots + k\left(k + 1\right) + \left(k + 1\right)\left(k + 2\right) = \frac{1}{3}\left(k + 1\right)\left(k + 2\right)\left(k + 3\right)$

Consider $\text{LHS}1.2 + 2.3 + 3.4 + \cdots + k\left(k + 1\right) + \left(k + 1\right)\left(k + 2\right)$

$= \frac{1}{3} k\left(k + 1\right)\left(k + 2\right) + \left(k + 1\right)\left(k + 2\right) = \frac{1}{3}\left(k + 1\right)\left(k + 2\right)\left(k + 3\right)$

$\text{LHS} = \text{RHS}$

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