Chapter 12, Problem 69RE

To determine

To prove the given formula for natural numbers using the mathematical induction.

Explanation of Solution

Given:

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

Proof:

For $n=1$ , left side is $\left(1+\frac{1}{1}\right)=\left(1+1\right)=2$ , and right side is $1+1=2$ . So, the given proposition is true for $n=1$ .

Let the proposition is true for $n=k$ .

Then, $\left(1+\frac{1}{1}\right)\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\cdot \cdot \cdot \left(1+\frac{1}{k}\right)=k+1$ .

Now,

  $\begin{array}{c}\left(1+\frac{1}{1}\right)\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\cdot \cdot \cdot \left(1+\frac{1}{k}\right)\left(1+\frac{1}{k+1}\right)=\left(k+1\right)\left(1+\frac{1}{k+1}\right)\\ =\left(k+1\right)\left(\frac{k+1+1}{k+1}\right)\\ =k+2\\ =\left(k+1\right)+1\end{array}$

This shows that the given proposition is true for $n=k+1$ if it is true for $n=k$ .

Thus, by the principle of the mathematical induction given formula is true.