If `A_1` is the area area bounded by `|x-a_i| + |y|=b_i, i in N` ,where `a_(i+1) =a_i+3/2 b_i and b_(i+1)=b_i/2,a_i=0 and b_i=32`, then

Options

A. `A_(3)=128`
B. `A_(3)=256`
C. `underset(nrarroo)limoverset(n)underset(i=1)SigmaA_(i)=(8)/(3)(32)^(2)`
D. `underset(nrarroo)limoverset(n)underset(i=1)SigmaA_(i)=(4)/(3)(16)^(2)`

Answer

A, C

Explanation

`a_(1)=0,b_(1)=32,a_(2)=a_(1)+(3)/(2)b_(1)=48,b_(2)=(b_(1))/(2)=16`
`a_(3)=48+(3)/(2)xx16=72,b_(3)=(16)/(2)=8`

So the three loops from i=1 to i=3 are alike.
Now area of ith loop (square) `=(1)/(2) ("diagonal")^(2)`
`A_(i)=(1)/(2)(2b_(i))^(2)=2(b_(i))^(2)`
`So, (A_(i)+1)/(A_(i))=(2(b_(i+1))^(2))/(2(b_(i))^(2))=(1)/(4).`
So the areas from a G.P. series.
So, the sum of the G.P. up to infinite terms is
`A_(1)(1)/(1-r)=2(32)^(2)xx(1)/(1-(1)/(4))`
`=2xx(32)^(2)xx(4)/(3)=(78)/(3)(32)^(2)` sq. units.