Disappearing Triangles

The perimeter of the largest triangle is $96$ cm. The perimeter of the second triangle is only half that of the first because all the triangles are equilateral and they are being separated at the midpoints.

Thus the answer is $96 + 49 + 24...$ .

This is an infinite geometric series. where the infinite sum is ${ S }_{ \infty }=\frac { { t }_{ 1 } }{ 1-r }$

This is then ${ S }_{ \infty }=\frac { 96 }{ 1-\quad (\frac { 1 }{ 2 } ) } =\frac { 96 }{ .5 } =96\times 2=192$

since $r=\frac { 1 }{ 2 }$ and ${ t }_{ 1 }=92$