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For any positive integers n > 1, let P(n) denote the largest prime not exceeding n. Let N (n) denotes the next prime larger than P(n). ( For example P(10) = 7, and N(10) = 11, while P(11) = 11, and N(11) = 13) If n + 1 is a prime number. Prove that the value of the sum

1/P(2)N(2) + 1/P(3)N(3) + 1/P(4)N(4) + ...... + 1/P(n)N(n) = (n - 1)/(2n + 2)

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Let p and q be two consecutive, p < q. If we take any n such that p ≤ n < q, we see that p(n) = p & N(n) = q. Hence the term 1/pq occurs in the sum q – p times.The contribution from such terms is (q - p)/pq = 1/p - 1/q since n+1 is prime, we obtain

Here p is used for the prime preceeding n + 1

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