Problem 4

Let $R$ be a ring with identity 1, and let $x\in R$.

1. Find four closed forms for the geometric series

   $$1+x+x^{2}+x^{3}+\cdots+x^{n}.$$

2. What are the condition(s) such that each of the closed forms are valid?

3. Factor 112 into primes, and thus evaluate 112 in $\mathbb{Z}_{37}$.

4. Hence, using the result(s) above, evaluate

   $$1+2^{3}+2^{6}+2^{9}+\cdots+2^{72}$$

   in the ring $\mathbb{Z}_{37}$.