Question

5: Legend has it that the game of chess was invented for the amusement of a Persian shah - or an Indian maharajah, or a Chinese emperor - who became so enthusiastic that he wanted to reward the inventor, who desired only one grain of wheat on the first square of the chessboard, two grains on the second square, four on the third, and so on, doubling the number of grains for each successive square on the 64-square chessboard. (From Mathematics: From the Birth of Numbers by Jan Gullberg.) How many grains of wheat is that? You can compute that value by computing the sum of the first 64 terms of this infinite series: 1 + 2 + 4 + 8 + 16 + 32 + 64 + ... In other words, the shah/maharajah/emperor owes the inventor of chess a mere 18,446,744,073,709,551,615 grains of wheat. Use Python and the amazing power of recursion to compute the sum of the first n terms of the series shown above. (There is a simple formula for computing this value that does not involve recursion -- if you know it, do not use it.) Call your function doubler. Write only one function. Do not use tail recursion. Your function should expect one argument, representing the number of terms of the series to be used in the computation. The argument passed to doubler will always be an integer greater than 0. Be sure to provide any needed type declarations. Here are some examples of how your function should work: >>> doubler(1) >>> doubler (2) 3 >>> doubler (3) >>> doubler (4) 15 >>> doubler (64) 18446744073709551615 Write your answer in the space below.

Answer 1

def doubler(n):
    if n == 1:
        return 1
    else:
        return 2*doubler(n-1) + 1


# Testing the function here. ignore/remove the code below if not required
print(doubler(1))
print(doubler(2))
print(doubler(3))
print(doubler(4))
print(doubler(64))

1 3 15 18446744073709551615