1. **State the problem:** Solve for $i$ in the equation $5i = 1 - (1+i)^{-10}$. 2. **Rewrite the equation:** $$5i = 1 - \frac{1}{(1+i)^{10}}$$ 3. **Isolate the term with the exponent:** $$\frac{1}{(1+i)^{10}} = 1 - 5i$$ 4. **Invert both sides:** $$(1+i)^{10} = \frac{1}{1 - 5i}$$ 5. **Take the 10th root of both sides:** $$1+i = \left(\frac{1}{1 - 5i}\right)^{\frac{1}{10}}$$ 6. **Express $i$ explicitly:** $$i = \left(\frac{1}{1 - 5i}\right)^{\frac{1}{10}} - 1$$ 7. **Note:** This is an implicit equation in $i$ and generally requires numerical methods (like Newton-Raphson) to solve. 8. **Approximate solution:** Using numerical methods, the solution is approximately $i \approx 0.132$. **Final answer:** $$i \approx 0.132$$