Do "inductively" and "recursively" have very similar meanings?

Question

Do "inductively" and "recursively" mean very similar?For example, if there is an algorithm that determines a n-dim vector by determine its first k+1 components based on its first k components having been determined, and is initialized with the first component, would you call it works recursively or inductively? I have been using "recursively", but today someone said it "inductively".

Answer 1

No they are not.

Inductively , means the statement is true for integers N=1,2,3,4,....

Therefore, upon proof, the statement holds true for ANY integer N.

Recursive definitions, functions, and sequences are defined in terms of themselves.

For example, the factorial. The factorial of a number is another factorial.

Specifically, n! = n*(n-1)! where 0! = 1

So a recursive function appears in it's own definition.

A recursive subroutine calls itself until a stopping condition is met.

So the above factorial function, for example, is implemented as:

factorial ( integer N)

{

if N>1 then factorial = N * factorial(N-1) <--- recursive call

else factorial = 1

}