An Indian folktale, recounted by Arab historian and geographer Ahmad al-Yaqubi in the 9th century, begins, "It is related by the wise men of India that when Husiya, the daughter of Balhait, was queen . . . ," and goes on to tell how the game of chess was invented. The queen was so delighted with the game that she told the inventor, "Ask what you will." The inventor asked for one grain of wheat on the first square of the chessboard, two grains on the second, four grains on the third, and so on, so that each square contained twice the number of grains as on the square before. (There are 64 squares on a chessboard.)

a. How many grains are needed
i. For the 8th square?
ii. For the 64th square?
iii. For the first row?
iv. To fill the board?
b. In sigma notation, write the series you used to fill the board.