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The thin lens formula is fundamental in optics and is used to determine the position and nature of images formed by lenses. It is given by the equation: \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \), where:

• \( f \) is the focal length of the lens.
• \( v \) is the image distance, from the lens to the image.
• \( u \) is the object distance, from the object to the lens.

This formula helps to calculate the image distance when the object's position and the focal length are known. In our given problem, the object (ant) is positioned 250 cm in front of the lens, which makes \( u = -250 \) cm (negative as per the sign convention). Given the focal length \( f = +50 \) cm, you can solve the formula to find the image distance, resulting in the first image forming 62.5 cm to the right of the lens. This image is real, as it forms on the opposite side of the lens from where the object is located.

Plane mirrors create images through reflection where the image distance is equal to the object distance from the mirror, but it appears on the opposite side. This is simple but vital in optics as it helps determine how images appear in everyday flat mirrors.
In this exercise, the image formed by the lens is reflected by the plane mirror placed 250 cm away from the lens. Initially, the image from the lens appears 187.5 cm to the left of the mirror, so the reflection's image forms 187.5 cm on the right in relation to the mirror. It results in a virtual image (I2) situated further in the positive direction relative to the original lens position, at 437.5 cm to the lens' right. Thus, even without moving, the plane mirror changes an incoming image’s direction.

Virtual images are those that cannot be captured on a screen because they appear to form inside a reflective object, like a mirror, rather than being projected like real images from a lens.
In our problem, after the image I1 is created by the lens and its reflection (I2) in the plane mirror, this reflected image behaves as a virtual object for the lens. This happens because image I2 appears behind the mirror and is used to predict where the new virtual image will form. As mentioned before, the virtual image (I2) is 437.5 cm from the lens, so when combined with the plane mirror, it helps create image I3 via the lens as another real image.

Solving optics problems often involves understanding various principles and sometimes combining them. These problems can include tasks such as determining image positions and characteristics using known parameters like distances and focal lengths. You must rely on:

• Drawing accurate ray diagrams to visualize light paths.
• Applying mathematical formulas like the thin lens equation.
• Using the concept of sign conventions, essential for understanding input values and calculating outcomes.

In this exercise, by sequentially using the thin lens formula and understanding mirror reflections, multiple images are successfully located. Breaking complex optical setups into simpler steps is a reliable way to untangle and solve even the trickiest problems in optics.