The Lombards press carrots by hand to make carrot juice. One fourth of the juice is extracted from the first pressing. Each subsequent pressing extracts one fourth of the remaining juice from the carrots. How many times will the carrots need to be pressed to extract at least two-thirds of the juice?
first press: 1/4 second press: 3/4* 1/4 = 3/16 third press: (3/4)^2* 1/4 = 3/4^3 then 1/4 + 3/4^2 + 3/4^3 + ... + 3/4^n 2/3 3/4^2 + ... + 3/4^n 5/12 1/4^2 + ... + 1/4^n 5/36 [1/16 - 1/4^(n+1)]/(3/4) 5/36 1/16 - 1/4^(n+1) 5/48 at this time ... no value of n will satisfy... 3/4^2 + 3/4^3 + ... + 3/4^n ... as n goes to infinity .. . . yields 1/4 adding the first value, 1/4 .. . .. . this adds up to at most 1/2 and thus will not yield the desired 2/3 level.
