Again Using Induction on N

Induction

Mathematical Induction

Subjects to be Learned

• first principle of mathematical induction
• basis footstep
• induction hypothesis
• induction
• second principle of mathematical induction

Contents

Commencement Priciple of Mathematical Induction

As we have seen in recursion , the set of natural numbers tin can be defined recursively, and its elements can be generated one past one starting with 0 by adding 1 . Thus the set of natural numbers can be described completely by specifying the basis element ( 0 ), and the process of generating an chemical element from a known element in the set.

Taking advantage of this, natural numbers can be proven to have certain backdrop as follows:
Beginning it is proven that the basis element, that is 0 , has the property in question (

basis pace ). Then it is proven that if an arbitrary natural number, announce information technology by n , has the belongings in question, and so the next element, that is n + 1 , has that property ( inductive step ).

When these ii are proven, then information technology follows that all the natural numbers have that holding. For since 0 has the property by the footing stride, the chemical element next to it, which is 1 , has the same property by the inductive pace. And then since i has the property, the element next to information technology, which is 2 , has the same property again past the anterior step. Proceeding likewise, whatsoever natural number tin be shown to have the property. This process is somewhat coordinating to the knocking over a row of dominos with knocking over the first domino respective to the basis step.

More more often than not mathematical statements involving a natural number northward such as 1 + 2 + ... + n = due north( due north + one )/2 can be proven by mathematical consecration by the aforementioned token.

To prove that a argument P ( due north ) is true for all natural number , where is a natural number, we continue as follows:
Basis Footstep: Show that P ( ) is true.
Induction: Evidence that for any integer , if P ( k ) is truthful (called

consecration hypothesis ), then P(yard+1) is true.
The first principle of mathematical consecration states that if the basis step and the inductive step are proven, then P ( n ) is truthful for all natural number .

Equally a first step for proof by induction,   it is often a good thought to recapitulate P(yard+1) in terms of P(k) so that P(k), which is assumed to be true, can exist used.

Instance:

Bear witness that for any natural number northward , 0 + 1 + ... + n = due north( n + 1 )/2 .

Proof:
Basis Step: If n = 0 , and then LHS = 0 , and RHS = 0 * (0 + ane) = 0 .
Hence LHS = RHS.
Induction: Assume that for an arbitrary natural number n , 0 + 1 + ... + n = n( due north + 1 )/two . --------

Induction Hypothesis
To prove this for northward+1, first try to express LHS for northward+1 in terms of LHS for n, and somehow employ the induction hypothesis.
Here allow united states of america try
LHS for n + 1 = 0 + 1 + ... + n + (n + 1) = (0 + 1 + ... + n) + (due north + ane) .
Using the induction hypothesis, the last expression can be rewritten as
north( n + ane )/2 + (n + ane) .
Factoring (n + i) out, we get
(n + 1)(n + 2) / 2 ,
which is equal to the RHS for due north+ane.

Thus LHS = RHS for n+1.

End of Proof.

More examples can be found

here

.

As well an

example

is given on how induction might be used to derive a new result.

Exam Your Understanding of Induction

Indicate which of the following statements are correct and which are not.
Click True or Faux , then Submit. At that place is 1 ready of questions.

Second Priciple of Mathematical Induction

At that place is another class of induction over the natural numbers based on the second principle of induction to bear witness assertions of the form x P(x) . This form of induction does not crave the basis step, and in the inductive pace P(n) is proved assuming P(chiliad) holds for all k < n . Sure problems can be proven more easily by using the second principle than the first principle considering P(yard) for all k < n tin can be used rather than just P(n - ane) to prove P(n).

Formally the

second principle of induction states that

      if n [ g [ chiliad < northward P(k) ] P(n) ] , then due north P(n) tin can be concluded.

Hither grand [ m < n P(k) ] is the induction hypothesis.

The reason that this principle holds is going to exist explained

later on subsequently a few examples of proof.

Example 1: Allow us prove the following equality using the second principle:
For whatever natural number north , ane + 3 + ... + ( 2n + 1 ) = ( due north + i )² .
Proof: Assume that 1 + 3 + ... + ( 2k + i ) = ( thou + 1 )² holds for all 1000 , grand < n .
Then 1 + three + ... + ( 2n + one ) = ( i + 3 + ... + ( 2n - one ) ) + ( 2n + i )
= n² + ( 2n + 1 ) = ( northward + 1 )² past the induction hypothesis.
Hence by the second principle of induction one + 3 + ... + ( 2n + 1 ) = ( n + 1 )² holds for all natural numbers.

Example two: Prove that for all positive integer n, i ( i! ) = ( n + 1 )! - 1
Proof: Assume that
1 * 1! + 2 * 2! + ... + g * k! = ( k + i )! - 1 for all chiliad , k < n .
And then 1 * i! + 2 * ii! + ... + ( n - i ) * ( n - 1 )! + n * north!
= due north! - 1 + due north * n! by the induction hypothesis.
= ( north + 1 )n! - one
Hence past the second principle of induction i ( i! ) = ( n + ane )! - 1 holds for all positive integers.

Example 3: Prove that whatever positive integer n > 1, tin be written as the product of prime number numbers.

Proof: Assume that for all positive integers thou, north > k > 1, 1000 can exist written as the production of prime numbers.
We are going to prove that n can be written equally the product of prime numbers.

Since n is an integer, it is either a prime number or not a prime number number. If n is a prime number, then information technology is the product of 1 , which is a prime number, and itself. Therefore the statement holds true.
If northward is not a prime, and so it is a product of two positive integers, say p and q . Since both p and q are smaller than n , by the induction hypothesis they can be written as the product of prime numbers (Note that this is not possible if the First Principle is being used). Hence northward tin can besides exist written as the product of prime numbers.

Test your understanding of 2d principle of induction :

Indicate which of the following statements are right and which are not.
Click True or False , and so Submit. In that location is one set of questions.

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Source: https://www.cs.odu.edu/~toida/nerzic/390teched/math/induction/induction.html

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