Question

Suppose you and two other mathematicians are milling around the food table before aseminar, eyeing the last four slices of cheese. How can you divide the four slices equallyamong the three of you?(a) Based on common sense, how much cheese should each of you get?5

(b) One of your mathematician friends suggests the following solution: “Let’s eachtake one of the slices of cheese, which leaves one slice for us to share. Instead ofcutting the remaining piece in thirds, let’s cut it into quarters, and each take aquarter of the leftover slice. We’re going to keep doing this, always cutting theremaining slice into quarters, taking a slice until there are only crumbs too smallto be of interest to anyone. Sound good?”i. How much cheese will you have accrued after the third iteration? Fourth?Write the amount of cheese you will get via this process as an infinite sumusing sigma notation.ii. Based on your answer in part (a) what can you say about the value of thesum in part i?

Answer 1

cut each cheese slice into three equal pieces. so for four cheese slices, we get 12 pieces.

distribute 12 pieces to 3 people which is 4 pieces to each mathematician.

a) each will get 4*1/3 = 4/3

b) so we are asked to cut the fourth cheese slice into quarters.

each mathematician already gets a cheese slice.

let the quantity of fourth cheese slice be 1

for first iteration

by cutting into quarters, each mathematician will get 1/4 and the remaining slice is also 1/4

for second iteration

cutting 1/4 in quarters, so each mathematician will get 1/16 and the remaining slice will be 1/16 = 1/4²

for nth iteration, we get 1/4ⁿ . this can be written as t_n

so we can write infinite sum as S_{n =}

after third iteration , we will get S₃ =

S₃ = 1+ 1/4 + 1/4² + 1/4³ = 1 + 0.25 + 0.0625 + 0.015625= 1.328125

similarly,

after fourth iteration we will get S₄ = 1 + 1/4 + 1/4² + 1/4³ + 1/4⁴

S₄ = 1 + 0.25+0.0625+0.015625 + 0.00390625 = 1.33203125

ii) lets caluculate for infinite sum. S_infinite = 1 + 1/4 + 1/4² + 1/4³ + 1/4⁴ + ......... ( infinite Geometric Progression)

S_infinite = 1/1-1/4 = 4/3 = 1.33333

which is equal to the answer in (a)