Sequence/series problem

[rir0302]

Two people are sharing a pie: person 1 eats half of the pie, person 2 eats half of the remaining pie, then person 1 eats half of the remaining pie, etc. How much of the pie does each person eat?

The halves would be the series 1/2, 1/4, 1/8, 1/16, 1/32,1/64..
But since we need to find how much each person eats it needs to be split into
Person 1: 1/2, 1/8, 1/32...
Person 2: 1/4, 1/16, 1/64...
So then we need to find the formulas for each series and then solve for it as the limit approaches infinity?

Am I on the right track?

 

[Steven G]

Yes, you are on the right track. Please post back with your solution.

Since you know the sum (1 for 1 pie) and all the numbers in the 2nd list is 1/2 of the numbers in 1st list that can help you figure out the answer as well and even quicker than other methods.

 

[rir0302]

Geometric series sum n=0 to infinity
Person 1: (1/2)(1/4)^n
Person 2: (1/4)(1/4)^n
Sum formula c/(1-r): Person 1 = 2/3, Person 2 = 1/3

 

[Steven G]

Geometric series sum n=0 to infinity
Person 1: (1/2)(1/4)^n
Person 2: (1/4)(1/4)^n
Sum formula c/(1-r): Person 1 = 2/3, Person 2 = 1/3

What you did is fine.
You could have thought of the solution as follows:
You know the sum is 1 (after all 1 whole pie will be eaten!). (1/2)(1/4)^n/(1/4)(1/4)^n = 2
So person 1 eats twice as much as person 2. You should see that the answer are 2/3 and 1/3.
If you did not, then I would then say Person 2 ate x of the pie and person 1 ate 2x of the pie. So x+2x=1 and x=1/3