if-i-had-a-grid-with-4-quadrants-and-i-shaded-a-quarter-of-it-and-chose-one-of-the-4-quadrants-to-form-a-new-grid-and-shaded-it-and-repeated-the-process-infinitly-what-is-the-amount-of-the-shaded-a

Question Number 4941 by madscientist last updated on 25/Mar/16

i f i h a d a g r i d w i t h 4 q u a d r a n t s a n d i s h a d e d a

q u a r t e r o f i t a n d c h o s e o n e o f t h e 4 q u a d r a n t s t o f o r m a n e w g r i d

a n d s h a d e d i t a n d r e p e a t e d t h e p r o c e s s i n f i n i t l y w h a t

i s t h e a m o u n t o f t h e s h a d e d a r e a ?

a n d w h a t f o r m u l a w i l l s o l v e i t ?

Answered by FilupSmith last updated on 25/Mar/16

A t f i r s t : \frac{1}{4} s h a d e d

t h e n a n o t h e r \frac{1}{8}

\dots

S = \underset{i = 1}{\sum^{\infty}} \frac{1}{4^{i}}, r a t i o = r = \frac{1}{4}, 1 s t t e r m = a = \frac{1}{4}

∴ S = \frac{a}{1 - r} = \frac{\frac{1}{4}}{1 - \frac{1}{4}} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{4} \times \frac{4}{3}

S = \frac{1}{3}