No bashing required

Geometry Level 5

A circle $C_0$ with radius 1 touches both the axes and line $L_1$ that passes through point $P(0,4)$ and cuts the $x$ -axis at $(x_1, 0)$ .

Another circle $C_1$ is drawn touching the $x$ -axis, line $L_1$ and another line $L_2$ that passes through point $P$ and cuts the $x$ -axis at $(x_2, 0)$ and this process is repeated $n$ times.

Find $\displaystyle \lim_{n \to \infty} \frac {x_n}{2^n}$ .

The answer is 2.

1 solution

Ajinkya Shivashankar
Feb 17, 2017

As my calculation was really large, just giving the basic steps here.

First , Consider the triangle $(x_n,0)$ , $(x_{n+1},0)$ and $P$ . It has an incircle with radius $=1$ . So using formula for inradius, we get a recurrence relation between $x_n$ and $x_{n+1}$ (After lots of calculations!) Getting its closed form, it is $x_n=2(2^n-2^{-n})$ And thus the answer for the limit is 2!

Is that $4x_{n+1} = 5x_n + 3 \sqrt{x_n^2 +4}$ ? I can't solve this, but the question says it is an integer, I guess isn't 1, so is 2.

Kelvin Hong - 3 years, 5 months ago

No bashing required

The answer is 2.

1 solution

0 pending reports