Let denotes the th prime and denotes the numbers of primes . Prove that for the holds : ( )

Solution :Processing by induction for ,we have = = 15 62.

So , this is valid for n = 6.

Now suppose for some n 6.

We will show that . That is ,

)

Since n 6 , then and .

Now , it is easy to say that )

Now suppose for some n 6.

We will show that . That is ,

)

Since n 6 , then and .

Now , it is easy to say that )

Now, guarantees a prime number between and for all Thus )and )are each atleast , which follows that )+ ).

Now observe that , = )+ )+.

Then

=

– + )- ) ( since ,)//That is .

So, we are done .

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