Problem of the week - Feb - 03 - 2012

[raghuram.konduri]

For all positive integers n and m, the function A(n) equals the following product:
(1 + 1/2 + 1/2^2)(1 + 1/3 + 1/3^2)(1 + 1/5 + 1/5^2)...(1 + 1/pn + 1/pn2), where pn is the nth smallest prime number, while B(m) equals the sum of the reciprocals of all the positive integers from 1 through m, inclusive. The largest reciprocal of an integer in the sum that B(25) represents that is NOT present in the distributed expansion of A(5) is

(A) 1/4
(B) 1/5
(C) 1/6
(D) 1/7
(E) 1/8

Here as per the given condition in the question a(5) should have the following expansion right? as n=5
(1 + 1/2 + 1/2^2)(1 + 1/3 + 1/3^2)(1 + 1/5 + 1/5^2)
Is the answer not 1/7 as we cannot get 1/7 as the product if any 2 terms? Please Help!

[stud.jatt]

For all positive integers n and m, the function A(n) equals the following product:
(1 + 1/2 + 1/2^2)(1 + 1/3 + 1/3^2)(1 + 1/5 + 1/5^2)...(1 + 1/pn + 1/pn2), where pn is the nth smallest prime number, while B(m) equals the sum of the reciprocals of all the positive integers from 1 through m, inclusive. The largest reciprocal of an integer in the sum that B(25) represents that is NOT present in the distributed expansion of A(5) is

(A) 1/4
(B) 1/5
(C) 1/6
(D) 1/7
(E) 1/8

Here as per the given condition in the question a(5) should have the following expansion right? as n=5
(1 + 1/2 + 1/2^2)(1 + 1/3 + 1/3^2)(1 + 1/5 + 1/5^2)
Is the answer not 1/7 as we cannot get 1/7 as the product if any 2 terms? Please Help!

The answer here is (E) 1/8

I think you erred in assuming A(5) = (1 + 1/2 + 1/2^2)(1 + 1/3 + 1/3^2)(1 + 1/5 + 1/5^2)

actually A(5) is the expansion upto the 5th prime number and would equal

(1 + 1/2 + 1/2^2)(1 + 1/3 + 1/3^2)(1 + 1/5 + 1/5^2)((1 + 1/7 + 1/7^2)(1 + 1/11 + 1/11^2)

1/4, 1/5 and 1/7 are already present in the above expansion, 1/6 will be the product when 1/2 is multiplied by 1/3. Only 1/8 won't be present as it requires the third power of 2 and here the highest power of 2 is only 1/(2^2) = 1/4. And since the rest of the numbers are primes we won't have another 2 to get 1/8

[tim]

thanks!