Making his way to infinity

Geometry Level 3

An insect starts moving from the origin, $(0,0)$ along the straight line in zig-zag manner. He first moves to $A = (3,3)$ , changes directions and walks half that distance to $B$ , changes direction and walks a further half of the distance to $C$ , so on and so forth.

If it ultimately ends at a point $(a,b)$ , then find the value of $a+b$ .

4 8 6 2

2 solutions

Caleb Hanger
Dec 9, 2015

x = 3 + 3/2 + 3/4 + 3/8 +...

x = 3 + x/2

x = 6

y = 3 - 3/2 + 3/4 - 3/8 +...

y = 3 - 3/2 + y/4

y = 2

So the insect ends up at (6,2). 6 + 2 = 8.

Luis Ortiz Montero
Dec 13, 2015

x-coordinate = a + a/2 + a/4 + a/8 ... + a/2^n = 2a

y-coordinate = a - a/2 + a/4 - a/8 ... + a/2^n = ?

x + y = 2a + 2a/4 + 2a/16 ... + 2a/4^n = 2a + ?

Using series we can get:

x + y = Σ (2a/4^n) , n from 0 to inf = 2a + ?

As 2a = 3 x 2 = 6

Then,

x + y = Σ (6/4^n) , n from 0 to inf

x+y = 6 Σ (1/4)^n , n from 0 to inf

x+y = 6 (4/3)

x + y = 8

Making his way to infinity

2 solutions

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