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1/(k+1)! to be calculated using tk . Terminate the loop when tk < . 1*10 13 . How
many iterations does it take?
E14. Write a loop (with initialization) that generates an approximation to pi, the
ratio of the diameter of a circle to its circumference. Do the work as in exercise
E13, but use this formula to calculate approximations:
pi = 4 - 4/3 + 4/5 - 4/7 + 4/9 - ...
Is this feasible? How many iterations does it take?
E15. Write a loop (with initialization) that generates an approximation to pi, ratio
of the diameter of a circle to its circumference. Do the work as in exercise E14,
but use this formula to calculate approximations, where c is 2*sqrt(3) :
pi = c - c/(3*3 1 ) + c/(5*3 2 ) - c/(7*3 3 ) + c/(9*3 4 ) - ...
Calculate c only once. Is this feasible —how many iterations does it take?
E16. Here is another way to calculate pi. Throw random darts at a disk of radius
1 that is inscribed in a 2 x 2 square. The fraction hitting the disk should be the ratio
of the area of the circle, to the area of the square: pi*r 2 / (2r) 2 , or pi/4 . To
throw a dart, calculate two random numbers (x, y) in the range -2..2 . The dart
hits the disk if x 2 +y 2 <= 1 . Write a loop (with initialization) that calculates an
approximation to pi by throwing random 10,000 darts.
E17. Write a loop to count how many times the vowel "a" occurs in a string s.
E18. Write a loop to count how many vowels a string s contains.
E19. Write a loop to count how many pairs of adjacent equal characters are in a
string s. The string "bbbccd" contains three pairs of adjacent equal values.

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