How Tall is an Infinite Stack of Russian Dolls?

[JuliusDarius]

Homework Statement

Your Russian friend gives you a housewarming gift, and you open it up to find a set of Russian Dolls. The first doll is 1 foot tall. Each time you open one of the wooden dolls you find a smaller doll inside it that is exactly 68.169% the size of its parent. Even when the dolls get very small, there is always another doll inside that is exactly 68.169% the size of the original. You decide to stack the Russian dolls on top of each other and measure how tall the entire set is.

When you add the heights of all the dolls up, you smile and recognize the number instantly!

How tall are the Russian Dolls when they are stacked up on top of each other (you can assume there are an infinite number of nested dolls)?

Homework Equations

Not Sure

The Attempt at a Solution

I assume the number would be infinity

 

[SammyS]

Homework Statement

Your Russian friend gives you a housewarming gift, and you open it up to find a set of Russian Dolls. The first doll is 1 foot tall. Each time you open one of the wooden dolls you find a smaller doll inside it that is exactly 68.169% the size of its parent. Even when the dolls get very small, there is always another doll inside that is exactly 68.169% the size of the original. You decide to stack the Russian dolls on top of each other and measure how tall the entire set is.

When you add the heights of all the dolls up, you smile and recognize the number instantly!

How tall are the Russian Dolls when they are stacked up on top of each other (you can assume there are an infinite number of nested dolls)?

Homework Equations

Not Sure

The Attempt at a Solution

I assume the number would be infinity

Not true.

The number of dolls is infinite, but the total height is not.

Write an expression for the height of the n^th doll.

 

[JuliusDarius]

Not true.

Write an expression for the height of the n^th doll.

Would it approach infinity, but never reach it?

 

[micromass]

Would it approach infinity, but never reach it?

No.

Write an expression for the nth doll

 

[JuliusDarius]

No.

Write an expression for the nth doll

Im not sure how to right an expression for the nth doll.

 

[micromass]

Write an expression for the first doll.
Write an expression for the second doll.
Write an expression for the third doll.
Write an expression for the fourth doll.
...

Notice a pattern.

 

[JuliusDarius]

write an expression for the first doll.
Write an expression for the second doll.
Write an expression for the third doll.
Write an expression for the fourth doll.
...

Notice a pattern.

1/68.169
2/68.169
3/68.169?

 

[JuliusDarius]

I'm not really sure what to write in this situation.

 

[Bohrok]

Each time you open one of the wooden dolls you find a smaller doll inside it that is exactly 68.169% the size of its parent.

What's 68.169% of 1 foot (the height of the first doll)? Then what's 68.169% of the height of this doll?