Use Principle of MI to verify (i) If n∈Z is a positive ineger then 2n3xn−1 divisible by 17. (ii) For all positive integers n≥5, 2k&gt;k2

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Use Principle of MI to verify  (i) If n∈Z is a positive ineger then 2n3xn−1 divisible by 17.  (ii) For all positive integers n≥5,  2k&amp;gt;k2

Sally Cresswell · Accepted Answer

According to the given information it is required to prove the following using principle of mathematical induction.  If n∈Z is a positive ineger then 2n32n−1 divisible by 17.  that is: 2n32n−1=xn9n−1=18n−1 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:  k∈Z is posisitve integer then 2k32k−1=18k−1 is divisible by 17  Now, prove that the statement is true for n = k+1:  So, consider  2n+132(n+1)−1=2n⋅2⋅32n−1  =18⋅2n32n−1  =17⋅2n32n+2n32n−1  Now, since  17⋅2n32n is divisible by 17 and from assumption 2n32n−1 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.

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

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