Question

A man places 1 penny on the square of a chess board, 2 pennies on the next square, 4 pennies on the third square, and continues this pattern. How many pennies will be on the 14th square?(1 point)
Responses

256 pennies
256 pennies

8,192 pennies
8,192 pennies

16,384 pennies
16,384 pennies

28 pennies

Answer 1

The pattern of placing pennies on the squares of the chessboard follows powers of 2. On the 1st square, there is \(2^0 = 1\) penny, on the 2nd square, there is \(2^1 = 2\) pennies, on the 3rd square, there is \(2^2 = 4\) pennies, and so on.

For the \(n\)-th square, the number of pennies is \(2^{n-1}\).

For the 14th square, we have:

\[
2^{14-1} = 2^{13}
\]

Calculating \(2^{13}\):

\[
2^{13} = 8192
\]

So, there will be **8,192 pennies** on the 14th square.