Question

Let $A(a,b)$ be a fixed point and $O$ be the origin of coordinates. If $A_1$ is the mid-point of $OA$, $A_2$ is the mid-point of $O{A}_1$ and so on. Then the coordinates of $A_n$ are:

• A. $\left(a(1-2^{-n}),b(1-2^{-n})\right)$
• B. $\left(a(2^{-n}-1),b(2^{-n}-1)\right)$
• C. $\left(a(1-2^{-(n-1)}),b(1-2^{-(n-1)}\right)$
• D. $(\frac{a}{2^n},\frac{b}{2^n})$

Answer 1

Answer: (D)

$(\frac{a}{2^n},\frac{b}{2^n})$

According to the problem,

Co-ordinates of $A_1$ are $(\frac{a}{2},\frac{b}{2})$.

Similarly co-ordinates of $A_2$ are $(\frac{a}{2^2},\frac{b}{2^2})$.

If we proceed in this way co-ordinates of $A_n$ are $(\frac{a}{2^n},\frac{b}{2^n})$.