Task B-4.2. The semi-minor axis, linear eccentricity, and semi-major axis of an ellipse are three consecutive terms of an arithmetic sequence. If the semi-major axis is increased by 1, the sequence will become geometric. Determine the equation of the ellipse.
## Solution.

Let $b, e, a$ form an arithmetic sequence. Then we have:

$$
\begin{aligned}
2 e & =a+b \\
e^{2} & =a^{2}-b^{2}
\end{aligned}
$$

From the first equation, we get $a=2 e-b$, and substituting this into the second equation yields

$$
e=\frac{4 b}{3} \text{ and } a=\frac{5 b}{3}
$$

Then $b, e, a+1$ form a geometric sequence, so we have

$$
e^{2}=b \cdot(a+1)
$$

or $\left(\frac{4 b}{3}\right)^{2}=b \cdot\left(\frac{5 b}{3}+1\right)$.

The semi-minor axis is $b=9$, and the semi-major axis is $a=15$. The equation of the ellipse is

$$
\frac{x^{2}}{225}+\frac{y^{2}}{81}=1
$$