Pleasez check the answers to these difficult questions?

[sapta]

A book of mine has the following questions with no answers.So I am not sure if I am right or if the proceure is correct and the best.So,please help-

1.For a given real number a>0,defineA_n=(1^a + 2^a + 3^a +...+n^a)^n
andB_n=n^n(n!)^a for all n=1,2,... Then
a. A_n<B_nfor all n>1
b.there exists an integer n>1 such that A_n<B_n
c.A_n>B_n for all n>1
d.there exists integers n and m both larger than one such that A_n>B_n and A_m<B_m

As there's no specification about the value of a in the options given,I considered a special case taking a=1 and then put n=2 and n=3.I found A_n>B_n and A_n<B_n respectively.So I think the answer is d.
2.Is 1 a prime number?

3.Let C denote the set of all complex numbers.Define A and B by
A={(z,w):z,w \inC and mod z=mod w}
B={(z,w):z,w \inC and z^2=w^2}

Then
a.A=B b.A\sqsubseteqB and A not equal to B
c.B\sqsubseteqA and B not equal to A
d.none of the above.

I think z^2=w^2 means mod z=mod w but the reverse is not true,so is the answer b?

4.If positive numbers a,b,c,d are such that 1/a,1/b,1/c,1/d are in A.P then we always have
a.a+d\geqb+c b.a+b\geqc+d
c.a+c\geqb+d d.none of the above

1/a +1/d =1/b +1/c or,(a+d)/ad =(b+c)/bc. Now 1/ad<1/bc or,ad>bc.so,a+d\geqb+c i.e.,(a)?

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thanking you in advance.And how do you put "not equal to" and "modulus" in latex?

 

[pizzasky]

Reply

Hi! Here are some of my thoughts with regard to your questions...

Q2) 1 is not a prime number. A prime number is defined as a number which is divisible only by itself and one. It can then be taken to mean that every prime number has 2 distinct factors. Hence, 1 is not prime.

Q3) Your reasoning that "z^2=w^2 means \mid z \mid =\ \mid w \mid but the reverse is not true" is valid. However, doesn't this mean that all elements in B are in A but not all elements in A are in B? So, the answer is...

Q4) (a) is indeed the correct option, but I do not understand one of the steps in your working. Why do you immediately conclude that \frac{1}{ad} &lt; \frac{1}{bc}? Can you clarify? Thanks.