**
Freshman Seminar 23j: Chess and Mathematics (Fall 2003)
**

Preliminary Question: Computing
The function

f(x) = 1 + x + x^{2} + x^{3} + x^{4}
+ ... + x^{2002} + x^{2003}
takes about 2 million arithmetic operations to compute as it stands,
using n-1 multiplications to compute each term x^{n}.
Rewriting it as
f(x)=(1-x^{2004})/(1-x)
still leaves more than 2000 operations -- one of which is a division,
which may be problematic
(harder to implement, and imprecise if x is too close to 1).
But the number of operations to compute f(x)
can be brought much lower than 2000. In how few operations
can you calculate f(x):
i) Using all four arithmetic operations +, - ,* , /
(but no logarithms and exponentials...),

ii) Using only additions and multiplications?