Modular Arithematic-see if my answer is correct

 P: 84 Q- What is the remainder when 1+2+2$$^{2}$$+...+2$$^{219}$$ is divided by 5. Solution: 2$$^{0}$$=1 mod5 2$$^{1}$$=2 mod5 2$$^{2}$$=4 mod5 2$$^{3}$$=3 mod5 2$$^{4}$$=1 mod5 Now I take (1,2,4,3) to be a set numbers. Since the summation goes to 219, there are a total of 220/4 = 55 sets. So I add 1+2+4+3= 10 and 10*55 = 550 <- my answer.
 Emeritus Sci Advisor PF Gold P: 11,155 Oops, I made a silly mistake. You are correct that the pattern 1,2,4,3 is repeating, since $2^{m+4} \equiv 2^m ~(mod 5)$. So you have found out that: $$\sum_{n=0}^{219}2^n \equiv 550~(mod 5)$$ What is the least residue of 550 (mod 5)? That is the required answer. Now alternatively, you should also be capable of identifying the kind of series that is given to you and evaluating it directly (before looking at congruences).
 P: 626 Geometric series? Spoiler Doesn't that sum to $$2^{220} - 1$$?