A spiral is made up of …

Let L₁, L₂, L₃, L₄, ...,L₁₃ be the lengths of semicircles of radii 0.5 cm, 1 cm, 1.5 cm, 2 cm,... and {tex}\frac{{13}}{2}{/tex}cm respectively.
Then, we have
L₁ = ({tex}\pi {/tex} {tex}\times{/tex} 0.5) cm = {tex}\frac{\pi }{2}{/tex}cm,
L₂ = ({tex}\pi {/tex} {tex}\times{/tex} 1) cm = 2 ({tex}\frac{\pi }{2}{/tex})cm, 
L₃ = ({tex}\pi{/tex} {tex}\times{/tex} 1.5)cm = 3({tex}\frac{\pi }{2}{/tex})cm,
L₄ = ({tex}\pi{/tex} {tex}\times{/tex} 2)cm = 4({tex}\frac{\pi }{2}{/tex})cm,....
L₁₃ =({tex}\pi{/tex} {tex}\times{/tex}{tex}\frac{{13}}{2}{/tex})cm = 3({tex}\frac{\pi }{2}{/tex})cm.
{tex}\therefore{/tex} total length of the spiral
= L₁ + L₂ + L₃ + L₄ +... +₁₃
={{tex}\frac{\pi }{2}{/tex}+2({tex}\frac{\pi }{2}{/tex})+3({tex}\frac{\pi }{2}{/tex})+4({tex}\frac{\pi }{2}{/tex})+....+13({tex}\frac{\pi }{2}{/tex})}cm
={tex}\frac{\pi }{2}{/tex}(1+2+3+4+...+13)cm
= {tex}\frac{\pi }{2}{/tex}{tex}\times{/tex}{tex}\frac{{13}}{2}{/tex}{tex}\times{/tex}(1+13)cm
=({tex}\frac{1}{2}{/tex}{tex}\times{/tex}{tex}\frac{{22}}{2}{/tex}{tex}\times{/tex}{tex}\frac{{13}}{2}{/tex}{tex}\times{/tex}14)cm=143cm.
Hence, the required length of the spiral is 143 cm.