Let A (a,b) be a fixed point and O be the origin of coordionates. If `A_(1)` is the mid-point of OA, `A_(2)` is the mid- poind of `A A_(1),A_(3)` is the mid-point of ` A A_(2)` and so on. Then the coordinates of `A_(n)` are

Options

A. `(a(1-2^(-n)),b(1-2^(-n)))`
B. `(a(2^(n-1)-1),b(2^(-n)-1))`
C. `(a(1-2^((n-1))),b(1-2^((n-1))))`
D. none of these

Answer

A

Explanation

The cordinates of `A_(1)` are `((a)/(2),(b)/(2))`
The corrdinates of `A_(2)` are `((a+(b)/(2))/(2),(b+(b)/(2))/(2))=((a)/(2)+(a)/(2^(2)),(b)/(2)+(b)/(2^(2)))`
the coordinates of `A_(3)` are `((a+(a)/(2)+(a)/(2^(2)))/(2),(b+(b)/(2)+(b)/(2^(2)))/(2))=((a)/(2)+(a)/(2^(2))+(a)/(2^(3)),(b)/(2)+(b)/(2^(2))+(b)/(2^(3)))`
Continunin in this manner we obscrve that the coordinates of `A_(n)` are `((a)/(2)+(a)/(2^(2))+(a)/(2^(3))+.....+(a)/(2^(n)),(b)/(2)+(b)/(2^(2))+(b)/(2^(3))+....+(b)/(2^(n)))`
`=(a(1-(1)/(2^(n))),b(1-(1)/(2^(n))))=(a(1-2^(-n)),b(1-2^(-n)))`