2012 UNCO Math Contest II Problems/Problem 9

Problem

Treasure Chest . You have a long row of boxes. The 1st box contains no coin. The next $2$ boxes each contain $1$ coin. The next $4$ boxes each contain $2$ coins. The next $8$ boxes each contain $3$ coins. And so on, so that there are $2^N$ boxes containing exactly $N$ coins.

(a) If you combine the coins from all the boxes that contain $1, 2, 3$, or $4$ coins you get $98$ coins. How many coins do you get when you combine the coins from all the boxes that contain $1, 2, 3,\ldots,$ or $N$ coins? Give a closed formula in terms of $N$. That is, give a formula that does not use ellipsis $(\ldots)$ or summation notation.

(b) Combine the coins from the first $K$ boxes. What is the smallest value of $K$ for which the total number of coins exceeds $20120$ ? (Remember to count the first box.)


Solution

(a) ${N-1}2^{N+1}+2$ (b) $2201$

See Also

2012 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
1 2 3 4 5 6 7 8 9 10
All UNCO Math Contest Problems and Solutions