Chapter 9.5, Problem 36E

Solution

A response to Kitty to clear up her confusion.

A response to Kitty to clear up her confusion has been determined.

Given:

Kitty is having trouble understanding mathematical induction proofs because she does not understand the inductive hypothesis. If we can assume it is true for $k$ , she asks, why can’t we assume it is true for $n$ and be done with it? After all, a variable is a variable.

Concept used:

The principle of mathematical induction states that if something is true for $1$ , then assuming it is true for $k$ , if it follows that it is true for $k+1$ , then it is true for all natural numbers $n$ .

Calculation:

The first step of mathematical induction is proving that the given statement is true for a base case (usually $1$ , sometimes $2$ ).

The second step of mathematical induction is assuming that the given statement is true for some intermediate number $k$ . It should be noted that the variable name does not matter. The variable to denote this intermediate number may as well be $n$ . However, in that case, the inductive hypothesis claims that the given statement is true for a specific number $n$ , not all natural numbers $n$ .

The third step of mathematical induction is proving that the given statement is true for $k+1$ , if the intermediate number was taken as $k$ or alternatively proving that the given statement is true for $n+1$ , if the intermediate number was taken as $n$ .

Only after completing all three steps, it will follow that the given statement must be true for all natural numbers.

Conclusion:

A response to Kitty to clear up her confusion has been determined.