Q. 170

# Long back in ancient times, a farmer saved the life of a king’s daughter. The king decided to reward the farmer with whatever he wished. The farmer, who was a chess champion, made an unusual request:“I would like you to place 1 rupee on the first square of my chessboard,2 rupees on the second square, 4 on the third square, 8 on the fourth square, and so on, until you have covered all 64 squares.Each square should have twice as many rupees as the previous square.” The king thought this to be too less and asked the farmer tothink of some better reward, but the farmer didn’t agree.How much money has the farmer earned?[Hint: The following table may help you. What is the first square on which the king will place at least Rs 10 lakh?]

A chess board consist of 8 by 8 squares.

It is evident from the problem that the arrangement is in the powers of 2.

The first square has 20 rupee coin.

The second square has 21 rupee coin.

The third square has 22 rupee coin, and so on.

Total sum S = 20 + 21 + 22 + 23 + 24 + … + 263 (64 terms)

This is in a series called a geometric progression, whose sum can be calculated using the formula S = where a is the first term, i.e 20 = 1, n is the number of terms and r is the common ratio which is 2.

S = = Rs. 3,68,93,48,81,47,41,91,03,230.

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