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The midpoints of the sides of a square of side 6 are joined to form a new square. This procedure is infinitely repeated for each new square, what is the sum of the areas of all the squares?
Just take the points, I already have the answer!

Respuesta :

Lets see

For side=6

  • Area=side²=6²=36units²

Now midpoints are the connections so diagonal of half side are the sides of new one

New side

  • √(6/2)2+(6/2)²
  • √3²+3²
  • √18
  • 3√2

New area

  • (3√2)²=18

Again new area will be half of old area

  • 18/2=9

Again

  • 9/2=4.5

This forms a geometric sequence

  • First term=a=36
  • Common ratio=r=(0.5)

Now

sum for infinite series:-

[tex]\\ \rm\Rrightarrow \dfrac{a}{1-r}[/tex]

[tex]\\ \rm\Rrightarrow \dfrac{36}{1-0.5}[/tex]

[tex]\\ \rm\Rrightarrow \dfrac{36}{0.5}[/tex]

[tex]\\ \rm\Rrightarrow 72units^2[/tex]

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