1. **State the problem:** We have a geometric sequence $\{G_n\}_{n=1}$ satisfying the relation $G_n + G_{n-1} = -1$ and the value $G_3 = 5$. We want to find the hundredth term $G_{100}$. 2. **Recall the definition of a geometric sequence:** Each term is obtained by multiplying the previous term by a common ratio $r$, so $G_n = G_1 r^{n-1}$. 3. **Use the given relation:** Since $G_n + G_{n-1} = -1$, substitute $G_n = G_1 r^{n-1}$ and $G_{n-1} = G_1 r^{n-2}$: $$G_1 r^{n-1} + G_1 r^{n-2} = -1$$ Divide both sides by $G_1 r^{n-2}$ (assuming $G_1 \neq 0$ and $r \neq 0$): $$r + 1 = \frac{-1}{G_1 r^{n-2}}$$ 4. **Note the left side $r+1$ is constant (independent of $n$), but the right side depends on $n$ unless $G_1 r^{n-2}$ is constant.** This is only possible if $G_1 r^{n-2}$ is constant for all $n$, which implies $r=1$. 5. **Check if $r=1$ satisfies the relation:** Then $G_n = G_1$ for all $n$, so $G_n + G_{n-1} = G_1 + G_1 = 2G_1 = -1$, so $G_1 = -\frac{1}{2}$. 6. **Check $G_3$ value:** $G_3 = G_1 r^{2} = -\frac{1}{2} \times 1 = -\frac{1}{2}$, but given $G_3 = 5$, so $r=1$ is invalid. 7. **Try to find $G_n$ explicitly:** The relation $G_n + G_{n-1} = -1$ can be rewritten as a non-homogeneous linear recurrence: $$G_n = -1 - G_{n-1}$$ 8. **Solve the recurrence:** The homogeneous part is $G_n^h = -G_{n-1}^h$, characteristic equation $r = -1$, so $G_n^h = A(-1)^n$. 9. **Find a particular solution $G_n^p = c$ constant:** Substitute into the recurrence: $$c = -1 - c \implies 2c = -1 \implies c = -\frac{1}{2}$$ 10. **General solution:** $$G_n = A(-1)^n - \frac{1}{2}$$ 11. **Use $G_3 = 5$ to find $A$:** $$5 = A(-1)^3 - \frac{1}{2} = -A - \frac{1}{2} \implies -A = 5 + \frac{1}{2} = \frac{11}{2} \implies A = -\frac{11}{2}$$ 12. **Final formula:** $$G_n = -\frac{11}{2} (-1)^n - \frac{1}{2}$$ 13. **Find $G_{100}$:** Since 100 is even, $(-1)^{100} = 1$, so $$G_{100} = -\frac{11}{2} \times 1 - \frac{1}{2} = -\frac{11}{2} - \frac{1}{2} = -\frac{12}{2} = -6$$ 14. **Check options:** None of the options (5, -5, 1, -1) equals -6, so re-examine the problem. 15. **Re-examining the problem statement:** It says $G_n + G_{n-1} = -1$ and $G_3 = 5$, but the sequence is geometric, so $G_n = G_1 r^{n-1}$. 16. **Rewrite the relation using geometric terms:** $$G_n + G_{n-1} = G_1 r^{n-1} + G_1 r^{n-2} = G_1 r^{n-2} (r + 1) = -1$$ 17. **For this to hold for all $n$, $G_1 r^{n-2} (r + 1)$ must be constant, so $r^{n-2}$ must be constant, implying $r=1$ or $G_1=0$ (not possible). So $r=1$. 18. **If $r=1$, then $G_n = G_1$ for all $n$, so $G_n + G_{n-1} = 2G_1 = -1$, so $G_1 = -\frac{1}{2}$. 19. **But $G_3 = G_1 = -\frac{1}{2}$ contradicts $G_3=5$. So no geometric sequence satisfies both conditions exactly. 20. **Conclusion:** The problem as stated is inconsistent for a geometric sequence. 21. **Check if the problem meant $G_n + G_{n-1} = -1$ only for $n=3$:** $$G_3 + G_2 = -1$$ 22. **Use $G_3 = G_1 r^{2} = 5$ and $G_2 = G_1 r$:** $$5 + G_1 r = -1 \implies G_1 r = -6$$ 23. **From $G_3 = 5$, $G_1 r^{2} = 5$ and $G_1 r = -6$, divide the two equations:** $$\frac{G_1 r^{2}}{G_1 r} = \frac{5}{-6} \implies r = -\frac{5}{6}$$ 24. **Find $G_1$ from $G_1 r = -6$:** $$G_1 \times \left(-\frac{5}{6}\right) = -6 \implies G_1 = \frac{-6}{-5/6} = -6 \times \frac{6}{-5} = \frac{36}{5} = 7.2$$ 25. **Find $G_{100}$:** $$G_{100} = G_1 r^{99} = \frac{36}{5} \times \left(-\frac{5}{6}\right)^{99}$$ 26. **Simplify $r^{99}$:** $$\left(-\frac{5}{6}\right)^{99} = (-1)^{99} \times \left(\frac{5}{6}\right)^{99} = - \left(\frac{5}{6}\right)^{99}$$ 27. **Calculate $G_{100}$:** $$G_{100} = \frac{36}{5} \times - \left(\frac{5}{6}\right)^{99} = - \frac{36}{5} \times \left(\frac{5}{6}\right)^{99}$$ 28. **Rewrite $\left(\frac{5}{6}\right)^{99}$ as $\frac{5^{99}}{6^{99}}$ and multiply by $\frac{36}{5} = \frac{6^2}{5}$:** $$G_{100} = - \frac{6^2}{5} \times \frac{5^{99}}{6^{99}} = - \frac{6^2}{5} \times \frac{5^{99}}{6^{99}} = - \frac{6^{2}}{6^{99}} \times \frac{5^{99}}{5} = - 6^{-97} \times 5^{98}$$ 29. **Since $6^{-97} = \frac{1}{6^{97}}$, the term is very small but negative. The exact value is complicated but the sign is negative and magnitude is very small. 30. **Among the options, the only negative values are -5 and -1. Since the term is very small in magnitude, closest is -1.** **Final answer:** D. -1