Simplify \(i^1+i^2+i^3+\cdots+ i^{97} + i^{98}+i^{99}.\)
\(i^{4k+2}+i^{4k+4}=0 \\ i^{4k+1}+i^{4k+3}=0\)
\(\text{we can rewrite the sum as}\\ (i + i^3) + (i^2 + i^4) + (i^5 + i^7) + (i^6 + i^8) + \dots + (i^{98}+i^{100}) +(i^{97}+ i^{99}) - i^{100} = -i^{100} \\ -i^{100} =- i^{4(25)} = -1\)
