The tail of a giant kangaroo is attached by a giant rubber band to a stake in the ground. A flea is sitting on top of the stake eyeing the kangaroo (hungrily). The kangaroo sees the flea leaps into the air and lands one mile from the stake (with its tail still attached to the stake by the rubber band). The flea does not give up the chase but leaps into the air and lands on the stretched rubber band one inch from the stake. The giant kangaroo, seeing this, again leaps
into the air and lands another mile from the stake (i.e., a total of two miles from the stake). The flea is undaunted and leaps into the air again, landing on the rubber band one inch further
along. Once again the giant kangaroo jumps another mile. The flea again leaps bravely into the air and lands another inch along the rubber band. If this continues indefinitely, will the flea
ever catch the kangaroo? (Assume the earth is flat and continues indefinitely in all directions.)
Okay so I tried this problem, and initially, I thought it was definitely a yes because every time the kangaroo jumps, the flea or whatever is dragged along with it. And eventually the flea will catch up the the kangaroo right? But then I tried it out in a Microsoft Excel document where
A is the #'s 1-100,
B is 5280*L1,
C is A/A*C+1 (i.e. C2 is A2/A1*C1+1 which is the ratio of the rubber band length before and after the kangaroo jumps [how much the flea is dragged along] plus the 1 inch that the flea jumps), and
D is B-C (for how far apart the flea and the kangaroo are)
The problem is, the difference between the flea and the kangaroo kept increasing and increasing so I have no idea what to do now. I know the answer is almost definitely yes... (otherwise what's the point of the problem?) but I can't get the math to work out! Is there a better way to prove it??
To analyze this problem mathematically, you can use the concept of infinite geometric series. Let's break down the problem step by step to find a solution.
1. Let's assume that the initial distance between the kangaroo and the flea is one mile. So, when the kangaroo jumps, it lands one mile away from the stake and the flea is one inch closer to the kangaroo.
2. After the kangaroo's first jump, the distance between them is 1 mile - 1 inch.
3. Now, for each subsequent jump, the flea lands one inch closer to the kangaroo, and the kangaroo jumps one mile further away. Therefore, after the kangaroo's second jump, the distance between them is (1 mile - 1 inch) + (1 mile - 1 inch) = 2 miles - 2 inches.
4. Following this pattern, after the kangaroo's third jump, the distance between them is (2 miles - 2 inches) + (1 mile - 1 inch) = 3 miles - 3 inches.
If we continue this process indefinitely, we can write the distance between the kangaroo and the flea after the nth jump as:
Distance_n = n miles - n inches.
Now, let's analyze the situation from the flea's perspective. Each jump of the flea is one inch closer to the kangaroo, which can be represented as:
Jump_n = 1 inch + (Jump_(n-1) + 1 inch).
So, the distance covered by the flea after the nth jump can be written as:
Distance_flea_n = 1 + (1 + (1 + ... (n-1) times) ).
We can simplify this equation using the formula for the sum of an arithmetic series:
Distance_flea_n = 1 + (1 + 2 + 3 + ... + (n-1) ).
The sum of integers from 1 to n-1 can be calculated using the formula for the sum of an arithmetic series:
Sum_n = (n-1) * (n-1 + 1) / 2 = (n-1) * n / 2.
So, we can rewrite the distance covered by the flea after the nth jump as:
Distance_flea_n = (n-1) * n / 2.
Now, let's compare the distances covered by the kangaroo and the flea:
Distance_flea_n = (n-1) * n / 2,
Distance_kangaroo_n = n miles - n inches.
If the flea catches the kangaroo, the distances covered by both of them should be equal. Therefore, we can set up the equation:
(n-1) * n / 2 = n - n.
Now, let's solve this equation to determine if there is a solution:
(n-1) * n / 2 = 0
n * (n - 1) = 0
n = 0 or n = 1.
This means that the flea can only catch the kangaroo if it jumps right at the beginning (n = 0) or if it jumps after the first jump of the kangaroo (n = 1).
However, in the given scenario, the flea never jumps at the beginning (n = 0), and it always jumps after the kangaroo's jump (n > 1). Therefore, the flea will never catch the kangaroo.
In conclusion, the flea will never catch the kangaroo in this scenario. The increasing distance between them is due to the fact that the kangaroo jumps one mile further away from the stake with each jump, outpacing the flea's shorter jumps.
Let the ratio of 1in/1mi be x
After each jump, we have the fraction of the distance of the flea from the roo:
1in/1mi
2in/2mi
3in/3mi
The flea is always 1/x of the way to the roo. He'll never make it.