Price of an item after 5 consecutive 10% discount.

Question

If an item costs $x$, What is the price of an item after $5$ consecutive $10$% discounts?

A common step-by-step calculation will take time.

After $1$st $10$% discount: $\frac{90x}{100}$.
After $2$nd $10$% discount: $\frac{81x}{100}$.
After $3$rd $10$% discount: ... ...

what is the faster (or, fastest) method to solve this?

Answer 1

After the $n$-th discount the price of $x$ will be $0.9^n x$. Hence, for $n=5$ you will have $$0.9^5 x = 0.59049 x.$$

Note that at $n=0$ (i.e. before any discount) we have $$0.9^0 x=1 \cdot x =x.$$

Answer 2

Use powers. Every $10\%$ discount reduces the price to $1-10\%=0.9$ of the preceding value, so:
after 1st discount $x_1 = 0.9\cdot x$,
after 2nd discount $x_2 = 0.9\cdot x_1 = 0.9^2\cdot x$,
after 3rd discount $x_3 = 0.9\cdot x_2 = 0.9^3\cdot x$,
.....
after Nth discount $x_N = 0.9\cdot x_{N-1} = 0.9^N\cdot x$.

For $N=5$ you have $$x_5=0.9^5\cdot x = 0.59049\,x$$