Let r: N a t-{0,1} →Rat be given by r(k)=∏i=2^k(1-(1)/(i)) . Write a recursive definition for r,

step 1 Starting with n upon k we can write it as n factorial upon k factorial n minus k factorial right

Let r: N a t-{0,1} →Rat be given by r(k)=∏i=2^k(1-(1)/(i)) ....

Let $r: N a t-\{0,1\} \rightarrow$ Rat be given by
$$
r(k)=\prod_{i=2}^k\left(1-\frac{1}{i}\right) .
$$
Write a recursive definition for $r$, and then prove that, for $k \geq 2$,
$$
r(k)=1 / k \text {. }
$$

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Key Concepts

Recursive Definition

A recursive definition specifies the value of a function for all inputs by providing a base case and then defining the function in terms of smaller or previous values. In problems like these, the function is initially defined for a minimal value, and subsequent values are computed by applying the recursive rule that relates each term to its predecessor.

Mathematical Induction

Mathematical induction is a proof technique used to prove statements or formulas that are indexed by the natural numbers. The method consists of proving the statement true for a starting value (base case) and then showing that if the statement holds for an arbitrary case k, it must hold for case k+1 (inductive step), thereby proving the statement for all natural numbers beyond the base case.

Product Notation and Telescoping Products

Product notation is used to denote the multiplication of a sequence of factors. In many instances, such products simplify via telescoping, where successive factors cancel out parts of previous ones, leading to a much simpler expression. Recognizing these patterns can simplify proofs and aid in verifying closed-form results.

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