Freshman Seminar 23j: Chess and Mathematics (Fall 2003)
Preliminary Question: Computing

The function

f(x) = 1 + x + x2 + x3 + x4 + ... + x2002 + x2003
takes about 2 million arithmetic operations to compute as it stands, using n-1 multiplications to compute each term xn. Rewriting it as   f(x)=(1-x2004)/(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?