I need a formula to find out number of circles required to reach a length of 100 million units in form of circle. All circles are in offset from first circle with +1 unit in radius. all circles have a common centre.
N=X*3.14*2 (x is the value of radius of circle)(N=perimeter of circle)
Y=N+6.28 ((Y is the value of circumference of next circle's radius with x+1unit) (a circle with a radius of 100 units has a perimeter of 628 units, a circle with 101 unit has a perimeter of 634.28 units, a circle with 102 unit has a perimeter of 640.56 units, the difference is always 6.28 units.))
so, addition of perimeter of first three circles is (628+634.28+640.56) 1902.84) units. what should be the formula to find out number of circles required to reach 100 million units.
We consider $n$ circles, where the radius of each circle is equal to $99+n$. The perimeter of the $n$th circle is therefore $$ p(n) = 2\pi (99+n) $$ When we add up the perimeters of the circles, we get the sum of an arithmetic progression $$ \begin{split} S =& p(1) + p(2) +p(3) +\ldots + p(n)\\ =& 2\pi \left(100+101+102 + \ldots + 99 +n \right)\\ =&2\pi(\underbrace{99+99+99 +\ldots+99}_{\text{n terms}}) + 2\pi \left(1+2+3+\ldots+n \right) \end{split} $$ The summing the $99$'s together makes just $99n$, and for the sum $1+2+\dots+n$ we can find the identity $$ 1+2+3+\ldots+n = \frac{n(n+1)}{2} $$ and therefore $$ S = 198\pi n + 2\pi \frac{n(n+1)}{2} $$ We want this to be at least $S= 10^8$, and therefore we get the equation $$ 2\times 10^8 = 396\pi n + 2\pi n^2 + 2\pi n \qquad \Rightarrow \qquad 2\pi n^2 + 398\pi n - 2\times 10^8 = 0 $$ Numerically, we can find $n\approx 5~544$. This is the number of circles required so that the sum of perimeters is at least to 100 million.