Tangent at a point $C_{1}$ (other than Origin $(0,0)$) on the curve $y=x^3$ meets the curve again at $C_{2}$. The tangent at $C_{2 }$ meets the curve at $C_{3}$ and so on. The abscissae of $C_{1},C_{2},C_{3},.......,{C}_{n}$ form a Geometric Progression .
Find the ratio of the area of the triangles $C_{1}C_{2}C_{3}$ and $C_{2}C_{3}C_{4.}$
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Select a suitable point $C_1$ and draw $T_1$, the tangent at it. Repeat the process.
Since the bases of the two triangles in question are the same $C_2C_3$, the required ratio is equal to $C_1Q : C_4P$ as shown.
Good luck.