MTH621 Assignment No: 1

 

MTH621 Assignment No: 1

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Question No: 1

Let suppose    is rational number.

Then   =p/q; (p,qϵ z , qo)

                           Here p and q are co-Prime number

     =     (p/q) → (A)

Taking square on both side

( ) ² =     (p/q) ²

18          =      p²/q²

18q²       =      p² → (1)

From eq (1) we see that

18 divided p²  is de    18  also divides p 

P             =        18x

1p           =        18x

Now we put the value of p in eq (1)

18q²        =          (18x) ²

18q²        =            (18)²x²

q²            =            18x² (2)

As 18 divides q² 

 18 divides   also   q

Hence from eq (1) and (2)

We can say that 18 is common factor 0f (p,q) , but we say that (p,q) are reletiveiiy prime .

And have no common factor which is contradiction to our supposition. Hence our supposition is wrong.  Thus is irrational number.

Question No: 2

Let 

We have to prove this by mathematically induction

Case (1)

For n=1

Put n=1 in eq (A) 

L.H.S     =     R.H.S

Thus the result is true for n=1

Case (2)

For n= k

Put n=k in eq (A)

Thus eq (A) will become  

 

Case (3)

Now

 Check it for n=k+1

So, we put n=k+1 in eq (B)