Use the Principle of Mathematical Induction to show that the...

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
\frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\cdots+\frac{1}{(2 n-1)(2 n+1)}=\frac{n}{2 n+1}
$$

Key Concepts

Principle of Mathematical Induction

Mathematical induction is a proof technique used to establish the validity of a statement for all natural numbers. It involves showing that the statement holds for an initial value (often n = 1) and then proving that if it holds for an arbitrary natural number n, it also holds for n + 1. This forms a chain of logical steps that extends the truth of the statement to all natural numbers.

Base Case

The base case is the first step in an inductive proof. It involves verifying that the statement is true for the initial value, typically the smallest natural number such as n = 1. This step is crucial because it provides the starting point for the induction and ensures that the subsequent inductive steps have a valid foundation.

Inductive Step

The inductive step is the phase in which the assertion is assumed to be true for an arbitrary natural number n, known as the induction hypothesis, and then shown to be true for n + 1. This step demonstrates that the truth of the statement for a single case implies the truth for the next case, effectively linking all natural numbers in a logical progression.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying mathematical expressions often used during the inductive step. This may involve factoring, combining like terms, or other methods of simplification to show that the step from n to n + 1 holds, thereby solidifying the inductive argument.

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