Bank Problems (Geometric Series)

A bank invites term deposits under the following conditions:
1. The deposit (or initial principal) may be any amount in dollars and cents, with a
minimum of $1.
2. The yearly interest, calculated at the end of each year and added to the principal, is
one cent less than 10% of the current principal, fractions of a cent being discarded.
3.The deposit with accumulated interest is returned at end of the sixth year.

Find the smallest initial deposit which would result in no fractions of cents being discarded
in any of the six years.

Hints.
Let $\displaystyle \large {\color{Yellow} x_{n}}$   be the value of the principal at the end of the nth year.
Derive a relationship between $\displaystyle \large {\color{Yellow} x_{n}}$   and $\displaystyle \large {\color{Yellow} x_{n-1}}$
Guess a relationship between $\displaystyle \large {\color{Yellow} x_{n}}$   and $\displaystyle \large {\color{Yellow} x_{0}}$   ( $\displaystyle \large {\color{Yellow} x_{0}}$   is the initial principal). This relationship should remind you of the compound interest formula.
Prove your guess by mathematical induction.
What's the sum of a geometric series?
Now think in terms of congruences.

Source: Lecture Notes on Mathematical Olympiad Courses: For Junior Section Vol 1 (Mathematical Olympiad Series)