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One side of an equilateral triangle is 18 cm. the mid point of its sides are joined to form another triangle whose mid point , in turn , are joined to form

another triangle. the process is continued indefinitely. find the sum of the perimeters of all triangles and areas of all triangle?



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Side length of first triangle = b
perimeter = 3b
side length of second triangle = b/2 , as it mid point
so perimeter = 3b/2
similarly, side length of third triangle = b/4
perimeter = 3b/4
and so on
sum of all perimeter will be 
3b + 3b/2 + 3b/4 + 3b/8 +......to ∞
it is a geometrical progression series.
sum of ∞ terms = a/(1 -r)
a= 3b = 54
r = 1/2
so sum = 54/(1 - 1/2) = 108 cm
now area,
area of first triangle = b²/4 ×(√3)
for second triangle = b²/16 ×(√3)
for third triangle = b²/64 ×(√3)
and so on
sum =  b²/4 ×(√3)+ b²/16 ×(√3) +  b²/64 ×(√3) + .... to ∞
this is also a GP with a = b²/4 ×(√3) and r =1/4
so sum = a/(1 -r) 
            = b²/4 ×(√3)/(1 -1/4) = 108√3 cm²
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