2nd Principle Of Mathematical Induction

Induction

Mathematical Induction -- Second Principle

Subjects to be Learned

second principle of mathematical induction

There is another form of induction over the natural numbers based on the second principle of induction to show assertions of the course

10 P(x) . This form of induction does non require the basis step, and in the anterior stride P(n) is proved bold P(grand) holds for all k < north . Certain problems can exist proven more hands by using the second principle than the commencement principle because P(k) for all thousand < n can be used rather than just P(n - ane) to evidence P(n).

Formally the

second principle of induction states that

if n [ k [ k < due north P(yard) ] P(northward) ] , and then due north P(n) can be concluded.

Here grand [ thousand < due north P(g) ] is the induction hypothesis.

The reason that this principle holds is going to be explained

later later a few examples of proof.

Instance ane: Let united states of america testify the following equality using the 2d principle:
For whatever natural number n , 1 + 3 + ... + ( 2n + 1 ) = ( northward + 1 )² .
Proof: Assume that one + 3 + ... + ( 2k + one ) = ( k + 1 )² holds for all k , chiliad < n .
Then ane + 3 + ... + ( 2n + 1 ) = ( ane + three + ... + ( 2n - 1 ) ) + ( 2n + 1 )
= nⁱⁱ + ( 2n + 1 ) = ( northward + 1 )² by the consecration hypothesis.
Hence past the 2nd principle of induction ane + 3 + ... + ( 2n + ane ) = ( northward + 1 )ⁱⁱ holds for all natural numbers.

Example 2: Testify that for all positive integer due north, i ( i! ) = ( northward + i )! - one
Proof: Assume that
1 * i! + 2 * ii! + ... + thou * thousand! = ( k + 1 )! - 1 for all k , k < n .
And so 1 * 1! + 2 * 2! + ... + ( northward - 1 ) * ( north - 1 )! + due north * n!
= north! - i + n * due north! by the induction hypothesis.
= ( n + ane )n! - one
Hence by the 2nd principle of induction i ( i! ) = ( n + ane )! - ane holds for all positive integers.

Example 3: Evidence that any positive integer n , north > one , tin can be written as the product of prime numbers.

Proof: Assume that for all positive integers g, n > 1000 > ane, k can be written as the product of prime numbers.
We are going to prove that n tin can be written as the product of prime numbers.

Since n is an integer, information technology is either a prime number or non a prime number. If n is a prime number number, then it is the product of one , which is a prime number number, and itself. Therefore the argument holds true.
If n is not a prime number number, then it is a product of two positive integers, say p and q . Since both p and q are smaller than north , by the induction hypothesis they can be written as the production of prime numbers (Note that this is non possible, or at least very hard, if the First Principle is existence used). Hence north can also be written every bit the production of prime numbers.

Exam your understanding of second principle of induction :

Indicate which of the following statements are correct and which are not.
Click True or Imitation , then Submit. There is one set of questions.

Side by side -- Introduction to Relation

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