Use Principle of MI to verify (i) If n in ZZ is a positive ineger then 2^n 3^(xn)-1 divisible by 17. (ii) For all positive integers n>=5, 2^k>k^2

hexacordoK 2020-12-27 Answered
Use Principle of MI to verify
(i) If nZ is a positive ineger then 2n3xn1 divisible by 17.
(ii) For all positive integers n5,
2k>k2
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Sally Cresswell
Answered 2020-12-28 Author has 91 answers
According to the given information it is required to prove the following using principle of mathematical induction.
If nZ is a positive ineger then 2n32n1 divisible by 17.
that is: 2n32n1=xn9n1=18n1 is divisible by 17
For n = 1
18-1=17
17 is diviseble by 17
so, the statement is true for n=1
Let us assume that the given statement is true for n = k that is:
kZ is posisitve integer then 2k32k1=18k1 is divisible by 17
Now, prove that the statement is true for n = k+1:
So, consider
2n+132(n+1)1=2n232n1
=182n32n1
=172n32n+2n32n1
Now, since
172n32n is divisible by 17 and from assumption 2n32n1 is divisible by 17
hence, 2n+132(n+1)1 is diviseble by 17
Therefore, whenever n = k is true n = k+1 is true.
Therefore, it is true for all integers.
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

New questions

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question