Determine all possible value(s) of a positive integer constant c that satisfy this relationship:

√(y-1) + √(y-2) + .....+ √(y-c) ≤ y

whenever y is a positive real number ≥ c

The line z = my touches the parabola z = sqrt(y - k) when the equation

m²y² = y - k has equal roots. i.e. when 4m²k = 1 giving m = 1/sqrt(4k).

Thus:    sqrt(y - k) <= y/sqrt(4k) for all values of y >= k.             (1)

Using k = 1 in (1) gives  sqrt(y - 1) <= y/2 < y
so the given relationship is true for c = 1.

Using k = 1 and k = 2 in (1) and adding the results gives:
sqrt(y - 1) + sqrt(y - 2) <= y/2 + y/sqrt(8) =< y
so the given relationship is true for c = 2.

When y = c + 1, the required relationship becomes

sqrt(1) + sqrt(2) + ..... + sqrt(c) <= c + 1

When c = 3 this becomes 1 + sqrt(2) + sqrt(3) <= 4 which is not true.

When c increases beyond 3, the amount, sqrt(c), added to the LHS is greater than the 1 added to the RHS, so the relationship is not true for c >= 3.

Posted by Harry on 2010-04-11 10:07:18