First slide

Cartesian plane

Question

Let $A (a, b)$ be a fixed point and $O$ be the origin of coordinates, If $A_{1}$ is the mid-point of $O A, A_{2}$ is the mid-point of -. $A_{1}, A_{3}$ is the mid-point of $A A_{2}$ and so on. Then the coordinates of $A_{1}$ are

Moderate

A

$(a (1 - 2^{- n}), b (1 - 2^{- n}))$
B

$(a (2^{- n} - 1), b (2^{- n} - 1))$
C

$(a (1 - 2^{- (n - 1)}), b (1 - 2^{- (n - 1)}))$
D

none of these

Solution

The coordinates of $A_{1}$ are $(\frac{a}{2}, \frac{b}{2})$
the coordinates of $A_{2}$ are $(\frac{a + \frac{a}{2}}{2}, \frac{b + \frac{b}{2}}{2}) = (\frac{a}{2} + \frac{a}{2^{2}}, \frac{b}{2} + \frac{b}{2^{2}})$
the coordinates of $A_{3}$ are
$\frac{a + \frac{a}{2} + \frac{a}{2^{2}}}{2}, \frac{b + \frac{b}{2} + \frac{b}{2^{2}}}{2}) = (\frac{a}{2} + \frac{a}{2^{2}} + \frac{a}{2^{3}}, \frac{b}{2} + \frac{b}{2^{2}} + \frac{b}{2^{3}})$
Continuing in this manner we observe that the coordinates of $A_{1}$ are
$\begin{aligned} (\frac{a}{2} + \frac{a}{2^{2}} + \frac{a}{2^{3}} + \dots \dots + \frac{a}{2^{n}}, \frac{b}{2} + \frac{b}{2^{2}} + \frac{b}{2^{3}} + \dots . + \frac{b}{2^{n}}) \\ = (a (1 - \frac{1}{2^{n}}), b (1 - \frac{1}{2^{n}})) = (a (1 - 2^{- n}), b (1 - 2^{- n})) \end{aligned}$

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