Semicircles and spiral

Simply the sum of 13 terms of the AP (0.5,1,1.5...)multiplied by 22/7(not pi).So we have

(22/7)*(13/2)(2*0.5+12*0.5)
=(22/7)*6.5*7
=22*6.5
=143

As we can see in the succession that the description of the problem presents, if the radii of a semicircle is \Nu , the radii of the next semicircle will be \Nu +0.5 cm Now, to know what is the measure of a semicirle is $\frac { 2\pi r }{ 2 } =\pi r$ Now making the sum of all the semicircles, will be $1\pi (0.5)+2\pi (0.5)+3\pi (0.5)+...+13\pi (0.5)$ = $\pi \left[ 1(0.5)+2(0.5)+3(0.5)+...+13(0.5) \right]$ = $\pi \left\{ 0.5\left[ 1+2+3+4+...+13 \right] \right\}$ = $\pi \left\{ 0.5\left[ \frac { 13(14) }{ 2 } \right] \right\}$ = $\frac { 22 }{ 7 } \left[ \frac { 1 }{ 2 } (\frac { 182 }{ 2 } ) \right]$ = $\frac { 4004 }{ 28 }$ = 143㎝

By using s=r x, where r is radius and x is the angle in radians Sum(s)= sum(r x) Since the angle is constant which is π.. Sum(s)=π*sum(r) By factorizing 0.5 Sum(s)=(1/2)(22/7)(1+2+...+13) (1+2+...+13)= (13)(14)/2=91 Sum(s)= (1/2)(22/7)(91)=143

All of the spiral spokes are semi circles. So the formula that we should be using to get the Length of a semi-circle (Spiral) is [Circumference of Circle (read Full Circle) / 2] i.e. (22/7*radius).

Now there are 13 semi circles with radii increment with 0.5 cm (i.e. 0.5, 1, 1.5 .... 6.5).

So the total length of all the semi-circles (spiral) is [(0.5 2/7) + 1 22/7 + 1.5 22/7 + ..... 6.5 22/7]. This now becomes simple AP by removing 0.5*22/7 (1+2+3+....13).

The answer is (0.5 22/7) * ((13 14)/2). Solving it as 143.