Chapter 12.4, Problem 35AYU

Challenge Problem Paper Creases If a sheet of paper is folded in half by folding the top edge down to the bottom edge, one crease will result. If the folded paper is folded in the same manner, the result is three creases. With each fold, the number of creases can be defined recursively by $c_1=1,c_{n+1}=2c_n+1$.

Find the number of creases for $n=3$ and $n=4$ folds.

Use the given information and your results from part (a) to find a formula for the number of creases after n folds, $c_n$, in terms of the number of folds alone.

Use the Principle of Mathematical Induction to prove that the formula found in part (b) is correct for all natural numbers.

Tosa Tengujo is reportedly the world’s thinnest paper with a thickness of $0.02mm$. If a piece of this paper could be folded $25$ times, how tall would the stack be?