Consecutive Percentage Increase-Decrease Problem

Question

If the price of sugar increases by 10% on the first day and decreases by 20% on the second day and increases by 30% on the third day and decreases by 40% on the fourth day and like this it continues on. On which day the price of the sugar will become 1.2 times of the original price?

I came across this Arithmetic question which I was not able to solve mathematically i.e. I was not able to prove mathematically that this is not possible. However I deduced this logically that this is not possible as the percentage by which the price falls is always greater than the percentage by which the price of the sugar increases and hence there will be no such day the price of the sugar will become 1.2 times of the original price?

Can someone please tell me that if there is a mathematical way possible to prove that there could be no such day possible?

Also what will happen if the question is flipped like on the first day the price decreases by 10% and the second day the price increases by 20% and on the third day the price decreases by 30% and on the fourth day it increases by 40% and like this it continues. Can we find the day when the price could be 'x' times of the original price?

Thanks!

Answer 1

$\delta = 10\% = \dfrac {1}{10}$

Then we need to find $n$ such that

$(1+\delta)(1-2\delta)(1+3\delta)\cdots (1\pm n\delta) = 1.2$

Note

$$(1+\delta)(1-2\delta) = 1 - \delta -2\delta^2 < 1$$

$$(1+3\delta)(1-4\delta) = 1 - \delta - 12\delta^2 < 1$$

$$(1+5\delta)(1-6\delta) = 1 - \delta - 30\delta^2 < 1$$

$$\vdots$$

$$(1+(2n-1)\delta)(1-2n\delta) = 1 - \delta - 2n(2n-1)\delta^2 < 1$$

Hence the price of sugar is, overall, decreasing. So it can never get to $1.2$.

Answer 2

On the tenth day the price of sugar decreases by $100\%$. So the sugar is worthless after day $10$, and thus you just need to check its value for the first nine days.

The prices are:

$$1 \to 1.1 \to 0.88 \to 1.144 \to 0.6864 \to 1.0296 \to 0.41184 \to 0.700128\to 0.1400256 \to 0.26604864 \to 0$$

so the price of $1.2$ is never reached.