Do you need multiple base cases for strong induction?

Strong induction is often used where there is a recurrence relation, i.e. an=an−1−an−2. In this situation, since 2 different steps are needed to work with the given formula, you need to have at least 2 base cases to avoid any holes in your proof.

Can you have two base cases in induction?

For application of induction to two-term recurrence sequences like the Fibonacci numbers, one typically needs two preceding cases, n = k and n = k − 1, in the induction step, and two base cases (e.g., n = 1 and n = 2) to get the induction going.

Can there be two base cases?

A recursive implementation may have more than one base case, or more than one recursive step. For example, the Fibonacci function has two base cases, n=0 and n=1.

What is the main reason someone would use the second principle of mathematical induction instead of the first principle?

Certain problems can be proven more easily by using the second principle than the first principle because P(k) for all k < n can be used rather than just P(n – 1) to prove P(n). The reason that this principle holds is going to be explained later after a few examples of proof.

Is there only one base case in recursion?

The base case is a simple case of the problem that we can answer directly; the base case does NOT use recursion. Each recursive algorithm must have at least one base case. Without at least one base case –> infinite recursion.

What are the two cases required in a recursive function?

So, to be a properly defined recursive function you must have a base case, i.e. a way for the function to return without making a recursive call, and your recursive calls must work towards the base case.

What are the principles of mathematical induction?

Mathematical induction is defined as a method, which is used to establish results for the natural numbers. Generally, this method is used to prove the statement or theorem is true for all natural numbers Write down the two steps involved in the principles of mathematical induction? Proving that the given statement holds true for the initial value.

Is 22n-1 is divisible by 3 using mathematical induction?

L.H.S. and R.H.S. are same. By mathematical induction, the statement is true. We see that the given statement is also true for n=k+1. Hence we can say that by the principle of mathematical induction this statement is valid for all natural numbers n. Show that 22n-1 is divisible by 3 using the principles of mathematical induction.

What is the induction step in the proof step?

In step 2, proving that the statement is true for the nth value, and also proving that true for the (n+1)th iteration also. This step is called the induction step

Why is the second principle called strong induction?

Note: The reason the second principle is called “strong induction” is that you can use it in other contexts to prove more than ordinary induction would. For example, imagine a situation in which you have a set that consists of two copies of the integers: a blue copy and a green copy, with the blue copy going “first”.