determine when π becomes greater than or equal to 3.14159265 and less than 3.14159266

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

CSC105 – Programming Logic

Lab04A:

Pi Approximation:  π can be computed using the following Nilakantha formula:

π  =  3 + 4/(2*3*4) - 4/(4*5*6) + 4/(6*7*8) - 4/(8*9*10) + 4/(10*11*12)…

or, where t = term, and i = iteration:

 

π  Value	=	3	3.1666666667		3.13333333333		3.14523809524		3.13968253968		Greater than or equal to 3.14 and less than 3.15
t	=		+	4/(2*3*4)	-	4/(4*5*6)	+	4/(6*7*8)	-	4/(8*9*10)	+	4/((2i)*(2*i+1)*(2*i+2))
i	=	0	1		2		3		4		i

 

Notice the pattern in the denominators above, and the sign alternates to its opposite (positive to negative, negative to positive, etc.).  This can be coded as a “for” loop where you initialize the variable “sign” as a positive 1 before the loop body, and then multiply it by negative 1 inside the loop to change the sign for each iteration of the loop.  Here is a graph of the π values after the first four iterations:

Create a Python program called “PiApproximation.py” that displays the following π value for i = 200, i = 500, and i = 1000.  Then, determine exactly which iteration causes the value of π to reflect 3.14159265…  In other words, determine when π becomes greater than or equal to 3.14159265 and less than 3.14159266.  Print this out at the end of your output (after the values of π are printed out at iterations 200, 500, and 1000).  All output should be done using the format() function, and decimals should go to the 10^th decimal place.

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