Calculus

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.

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  1. 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

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  2. 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

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

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