I have $100$ dollars at start.
I invest $50 \%$ of what I have after every trade I make($50$ dollars in 1st trade and then compounding).
Every successful trade gives me $85 \%$ profit of the money invested.
Every loser trade deducts the entire money invested($100 \%$ loss of the money invested in each trade).
Total Win-rate(profitable trades) is $70 \%$.
After how many trades will I $100 \times$ my initial investment($ \$ 100 $) or is it mathematically not possible?
We can think about this in terms of expected values, but there's no guarantee of where you'll end up when - that's what computer simulations are for.
Let's start with an initial wealth of $w_0$. Then, we invest half of it, and if the trade is profitable (with probability $0.7$), we end up with an $85 \%$ profit on the investment, i.e., the investment grows from $0.5w_0$ to $0.5w_0*1.85$. So, we end up with $0.5w_0 + 0.5w_0*1.85 = 1.425w_0$.
If the trade is not profitable (probability $0.3$), the investment goes from $0.5w_0$ to $0$, and we end up with $0.5w_0$. So, our wealth after that first trade will be:
$$ w_1 = \begin{cases} 1.425w_0 , & p = 0.7 \\ 0.5w_0 , & p = 0.3 \end{cases} . $$
The expected value of this trade is
$$ E[w_1] = 0.7*1.425w_0+0.3*0.5w_0 $$ $$ = 1.1475w_0 .$$
So, if you do this trade an infinite number of times, you can expect the long-run average growth rate in your wealth to be $14.75 \%$ per trade.
If you conduct this trade $n$ times, that gives you an expected wealth of $w_n = 1.1475^n w_0$, so you end up getting to $100$ times your initial wealth when:
$$ 1.1475^n w_0 = 100w_0 $$ $$ \Rightarrow 1.1475^n = 100 $$ $$ \Rightarrow n = \log_{1.1475} {(100)} $$ $$ \approx 33.47 . $$
So, on average, if you started with an infinite number of ensembles of wealth $w_0$ and ran this strategy, your 34th trade is the one that breaks 100 times initial wealth.
In reality, if you do this once, you could lose 34 times in a row, and end up at $0.5^{34}*w_0 \approx 5.8*10^{-11}$ of your initial wealth on trade 34. Or win 34 times in a row and end up at $ 1.425^{34}\approx 169709 $ times your initial wealth on trade 34. Or just about anywhere in between.