For i, k ∈N a t, define r(k, i)=[ if k=0 then 0 else r(k-1, i)+i] . Prove that, for any i,

step 1 If f of x equals x to the negative n, we want to prove the general formula. So that's going to b

For i, k ∈N a t, define r(k, i)=[ if k=0 then 0 else ...

For $i, k \in N a t$, define $$ r(k, i)=[\text { if } k=0 \text { then } 0 \text { else } r(k-1, i)+i] . $$ Prove that, for any $i, k \in N a t, r(i, k)=k i$. The proof should be based on the material in Section 3.2. [Caution: Be sure to keep track of the uses of $i$ and $k$ in the definition and in the assertion to be proved.]

For $i, k \in N a t$, define
$$
r(k, i)=[\text { if } k=0 \text { then } 0 \text { else } r(k-1, i)+i] .
$$
Prove that, for any $i, k \in N a t, r(i, k)=k i$. The proof should be based on the material in Section 3.2. [Caution: Be sure to keep track of the uses of $i$ and $k$ in the definition and in the assertion to be proved.]

Key Concepts

Natural Numbers

Natural numbers are the set of non-negative integers typically used for counting and ordering. They form the domain over which recursive functions are often defined, and proofs by induction are based on the well-ordered structure of natural numbers, which guarantees that every non-empty subset has a least element.

Inductive Step

The inductive step in the process of mathematical induction involves assuming that a statement holds for a particular case and then proving that the statement also holds for the next case. This step is crucial because it allows the property to be extended from the base case to all natural numbers, ensuring the completeness of the induction argument.

Recursion

Recursion is the process of defining a function in terms of itself. A recursive definition includes a base case, which establishes the value of the function for an initial input, and a recursive step that relates the function at one input to its value at a smaller or simpler input. This method is used to build complex operations, such as multiplication, from simpler ones like addition.

Mathematical Induction

Mathematical induction is a proof technique used to demonstrate that a property holds for all natural numbers. It involves first proving a base case, where the property is shown to be true for an initial number, and then proving that if the property holds for an arbitrary number, it also holds for its successor. This method is particularly useful for proving statements about recursively defined functions.

Base Case

The base case is the component of a recursive definition or an inductive proof that provides the starting point for the process. It specifies the value of a function or the validity of a statement for the initial element of a domain and ensures that the subsequent steps in the recursion or induction have a solid foundation.

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