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,define$$A_n=(1^a + 2^a + 3^a +...+n^a)^n$$
and$$B_n=n^n(n!)^a$$ for all n=1,2,... Then
a. $$A_n<B_n$$for 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 $$\in$$C and mod z=mod w}
B={(z,w):z,w $$\in$$C and $$z^2=w^2$$}

Then
a.A=B b.A$$\sqsubseteq$$B and A not equal to B
c.B$$\sqsubseteq$$A 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$$\geq$$b+c b.a+b$$\geq$$c+d
c.a+c$$\geq$$b+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$$\geq$$b+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 $$\frac{1}{ad} < \frac{1}{bc}$$? Can you clarify? Thanks.