Evaluate
\[i^{14762} + i^{14763} + i^{14764} + i^{14765}.\]
So I can find the remainder when I divide each exponent by 4, and that will tell me which value of $i$ to use. For example, $14762 \div 4 = 3690$ with a remainder of 2, so $i^{14762} = i^2 = -1$. Similarly, $14763 \div 4 = 3690$ with a remainder of 3, so $i^{14763} = i^3 = -i$. Therefore, the expression becomes \[(-1) + (-i) + (1) + (i).\] The real parts are $-1$ and $1$, which cancel out, and the imaginary parts are $-i$ and $i$, which also cancel out. So the expression simplifies to $0$.

