Arope is swinging in such a way that the length of the arc traced by a knot at its bottom end is decreasing geometrically. if the third arc is 20ft and the seventh arc is 12 ft. what is the length of the arc on the sixth swing?

Given that the length of the arc traced by a knot at its bottom end is decreasing geometrically
it means the difference between the length of every arc is the same 
and we know that
the third arc is 20 ft and the seventh arc is 12 ft
length of the sixth arc = ?
third arc = 20ft
fourth arc = 18ft
fifth arc = 16 ft
sixth arc = 14 ft
seventh arc = 12ft
Now you see that the difference is same and it is decreasing geometrically.
Thus the length of the sixth arc is 14 ft.

approximately 13.63 ft. Since we have a geometric series, the basic function for the arc length is: f(x) = N*R^x where N = constant value for initial amplitude R = Ratio of successive swings x = swing number Since we've been given the lengths of 2 different swings, let's express them as equations: 12 = N*R^7 20 = N*R^3 Now let's divide the 1st equation by the second, getting 12/20 = (N*R^7)/(N*R^3) The N terms can be canceled, both top and bottom, giving 12/20 = R^7/R^3 And we can simply subtract the exponents to perform the division. So we have 12/20 = R^4 0.6 = R^4 0.880111737 = R Now let's calculate N. 20 = N*R^3 20 = N*0.880111737^3 20 = N * 0.68173162 29.33705789 = N So our function is: f(x) = 29.33705789 * 0.880111737^x Let's verify for the given values of 3 and 7. f(3) = 29.33705789 * 0.880111737^3 f(3) = 29.33705789 * 0.68173162 f(3) = 20 f(7) = 29.33705789 * 0.880111737^7 f(7) = 29.33705789 * 0.409038972 f(7) = 12 Now let's calculate f(6) f(6) = 29.33705789 * 0.880111737^6 f(6) = 29.33705789 * 0.464758002 f(6) = 13.6346324 So the length of the arc on the sixth swing is approximately 13.63 ft.