Question

In a famous story, a wise man in India requests a reward promised by his king. The wise man shows the king a chessboard having 64 squares. "Put 1 grain of rice on the first square, 2 grains on the second square, 4 on the third square, 8 on the fourth square, and so on. That will be my reward."

The king readily agreed because this does not seen to be a very large request, but before he gets to the 64th square, he realizes he is ruined. What is the total amount of rice grains that the king must give the wise man at the 62nd square?

A. 2^30-1
B. 2^62-1
C. 2^31-1
D. 2^62
E. 2^32

Answer 1

one way to answer this question is look at a simpler problem
how many grains on 3 squares?
Put 1 grain of rice on the first square, 2 grains on the second square, 4 on the third square,
The sum of just 3 squares is
1+2+4 = 7= 8 -1

if we powers of 2, we could write this as
\[ 2^0 + 2^1 + 2^2 = 2^3 - 1 \]
if we do 4 squares:
\[ 1+2+4 + 8 = 15= 16 -1\\
2^0 + 2^1 + 2^2 +2^3 = 2^4 - 1 \]

do you see the pattern ?

or you could use the sum of geometric series, if you know that formula