Cover up some fractions!
1 1 = 2 1 + 2 1 , 2 1 = 3 1 + 6 1 , 3 1 = 4 1 + 1 2 1
Is it true that for all positive integers N , the value of N 1 can be expressed as the sum of the reciprocals of 2 positive integers: N 1 = A 1 + B 1 ?
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1 solution
If A and B don't have to be distinct (which is not a requirement here), then there is an even more trivial solution: A = B = 2N
2 N 1 + 2 N 1 = 2 N 2 = N 1
Yes, because for every positive integer N , at least one solution for N 1 = A 1 + B 1 is A = N + 1 and B = N ( N + 1 ) , because A 1 + B 1 = N + 1 1 + N ( N + 1 ) 1 = N ( N + 1 ) N + N ( N + 1 ) 1 = N ( N + 1 ) N + 1 = N 1 .