Use mathematical induction to prove the formula for all...

Use mathematical induction to prove the formula for all integers $n \geq 1$.
$$\left(1+\frac{1}{1}\right)\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right) \cdots\left(1+\frac{1}{n}\right)=n+1$$

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite sequence of statements. It involves two main steps: proving the base case and then proving that if the statement holds for an arbitrary case, it also holds for the next case. This method is particularly powerful for propositions indexed by the natural numbers.

Base Case

The base case is the first step in a proof by induction where the proposition is verified for the initial value, typically n = 1. Establishing the base case ensures that the property holds at the starting point, which is essential for the induction process to begin.

Inductive Hypothesis

The inductive hypothesis is an assumption made during the induction proof that the proposition holds true for some arbitrary integer n. This hypothesis is used as a stepping stone to prove that if the statement is true for n, then it must be true for n + 1.

Inductive Step

The inductive step is the part of an induction proof where, under the assumption of the inductive hypothesis, one shows that the statement holds for the next integer, n + 1. This step, combined with the base case, completes the proof by confirming the proposition for all integers n greater than or equal to the starting value.

Algebraic Simplification

Algebraic simplification involves manipulating expressions, such as products and sums, to reduce them to a form that is easier to analyze or compare. In the context of a proof by induction, it often helps in showing that the expression for n + 1 can be derived from the expression for n, thereby solidifying the inductive step.

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