1.prove 1+1/2+1/4+....+1/2(pi)=2-(1/2(pi)) for all natural numbers n 2.Prove tha

Question

1.prove 1+1/2+1/4+....+1/2(pi)=2-(1/2(pi)) for all natural numbers n

2.Prove that 7^n-6n-1 is divisible by 36 for all positive integers n

3. The principle of mathematical induction can be extended as follows. A list Pm, Pm+1,.....of propositions is true provided (i) Pm is true, (ii) Pn+1 is true whenever Pn is true and n is greater than or equal to m.

(a) Prove that n1>n+1 for all intgers n greater than or equal to 2

(b) Prove that n! > n^2 for all integers n greater than or equall to 4 [Recall that n!=n(n-1)...2.1; for example 5!=5*4*3*2*1=120]

4.For ech nsubset of natural numbers, let Pn denote the assertion "n^2+5n+1 is an even integar"

(a) Prove that Pn+1 is true whenever Pn is true

(b) For which n is Pn actually true? What is the moral of this excercise?

Explanation / Answer

2)

7^n - 1 = 6(7^(n-1)+7^(n-2)+...+1) so

7^n - 1 -6n = 6(7^(n-1)+7^(n-2)+...+1 -n)
= 6(7^(n-1)-1 + 7^(n-2)-1 +...+7-1 +1-1)

they are of form

7^k-1 for k in N thus divisible by 6

Therefore divisible by 36

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