# Find value of equation given

Given $$i+i^2+i^3+...+i^{2012}$$ find value of this. I do not know how to find value given so much numbers.I would like any hint given to solve these kind of problems.

• Hint: $\;1+x+x^2+\cdots+x^{2011} = (1-x^{2012}) / (1-x)$ – dxiv Jun 22 '17 at 17:38
• Add some of the first terms together. Can you find a pattern? Always try to find a pattern in scary questions like these. – greenturtle3141 Jun 22 '17 at 17:42
• @greenturtle3141 since the first 4 terms equal 0 , the answer will be 0 since the values are repeated – trying to learn Jun 22 '17 at 17:48

Hint:

$$i+i^2+i^3+i^4=i+(-1)+(-i)+1=0$$

Try to figure out what is

$$i^5+i^6+i^7+i^8.$$

• Well since i^5=i, i^6=-1 and so on , the answer will be 0 I guess – trying to learn Jun 22 '17 at 17:45
• yes, answer is $0$. Try to use geometric series approach as suggested by dxiv as well. It's good to tackle a problem using multiple approaches. – Siong Thye Goh Jun 22 '17 at 17:48

$$i+i^2+\cdots+i^{2012}=x \stackrel{×i} \Rightarrow$$

$$i^2+i^3+\cdots+i^{2012}+i^{2013}=xi \stackrel{(1)-(2)} \Rightarrow$$

$$i-i^{2013}=x(1-i) \Rightarrow x=\frac{i-(i^2)^{1006}\cdot i}{1-i}=0.$$