Legend has it that long ago a kind was very pleased with the game of chess that he decided to reward the inventor of the game Anna, with whatever she wanted. Anna asked for a resource instead of money. Specifically, she asked for one grain of wheat for the first square of a chessboard, two grains of wheat for the second square, four grains of wheat for the third square, and so on until the entire chess board was full. There are 64 squares on a chess board)

What would be an expression for the total amount of grain needed to fulfill Anna's request.

clearly

1+2+4+8+ ... + for 64 terms
a GS, where a=1, r=2 and n=64

sum(64) = 1(2^64 - 1)/(2-1)
= 2^64 - 1

notice for this series
sum(2) = 3 = 2^2 - 1
sum(3) = 7 = 2^3 - 1
sum(4) = 15 = 2^4 - 1
...
sum(64) = 2^64 - 1

2^0 = 1

2^1 = 2
2^2 = 4
2^3 = 8
say you had three squares. labeled 1, 2 and 3
the first square has 2^0 = 1
the second square has 2^1 = 2
the third square has 2^2 = 4
now go on to 64 th square has 2^63
now 2^63 = approximately 9.33 * 10^18

I gave you the amount on the last square. Use Reiny's system.

To find an expression for the total amount of grain needed to fulfill Anna's request, we can start by analyzing the pattern of doubling grains for each square.

We know that for the first square, Anna asked for one grain of wheat. This can be represented as 2^0, where the exponent 0 indicates the position of the square (zero-based indexing).

For the second square, Anna asked for two grains of wheat. This can be represented as 2^1, where the exponent 1 indicates the position of the square.

For the third square, Anna asked for four grains of wheat. This can be represented as 2^2.

We can observe the pattern that the amount of wheat for each subsequent square is obtained by doubling the previous square. So, for the fourth square: 2^3 grains, fifth square: 2^4 grains, and so on.

Therefore, the expression for the amount of grain needed for each square can be written as 2^(n-1), where n represents the position of the square.

To find the total amount of grain needed to fulfill Anna's request, we need to sum up the grains for all 64 squares. We can write this as:

2^0 + 2^1 + 2^2 + ... + 2^(n-1)

But since we know that there are 64 squares on a chessboard, we can substitute n with 64 in the expression:

2^0 + 2^1 + 2^2 + ... + 2^(64-1)

Simplifying this expression will yield the total amount of grain needed to fulfill Anna's request.