1. ## Three students and bread

Three students share a bread. The first eats half, the second eats half of what is left, and the third eats half of that and so on....How much does each eat?

2. Originally Posted by zhupolongjoe
Three students share a bread. The first eats half, the second eats half of what is left, and the third eats half of that and so on....How much does each eat?
student 1 = $\displaystyle \frac{1}{2}+\frac{1}{16}+\frac{1}{64}+...=\frac{1} {2^1}+\frac{1}{2^4}+\frac{1}{2^7}+...=\sum_{n=0}^{ \infty}\frac{1}{2^{3n+1}}=$

$\displaystyle \frac{1}{2}\sum_{n=0}^{\infty}\frac{1}{8^n}=\frac{ 1}{2}\left( \frac{1}{1-\frac{1}{8}}\right)=\frac{1}{2} \left( \frac{1}{\frac{7}{8}}\right)=\frac{1}{2} \cdot \frac{8}{7}=\frac{4}{7}$

You will find a similar pattern for students 2 and 3.

3. The fractions consumed by each

1 are 1/2 , 1/16 1/128 totalling ˝[1+(1/8) + (1/8)2 +…]

2 are 1/4 , 1/32, 1/256 totalling 1/4[1+(1/8) + (1/8)2 +…]

3 are 1/8, 1/64 1/512, totalling 1/8[1+(1/8) + (1/8)2 +…]

[1+(1/8) + (1/8)2 +…] = 1 / (1-1/8) = (8/7). Therefore,

1 consumes (1/2) (8/7) = 4/7

2 consumes (1/4) (8/7) = 2/7

3 consumes (1/8) (8/7) = 1/7

Note 4/7 + 2/7 + 1/7 = 1