ICSE Class 9 Answered

The second image will appear a distance 3d away from the observer.  One can continue this chain of reasoning to argue that images will appear facing the observer at d, 3d, 5d, 7d, and so on, in principle to infinity.

In order to calculate the total area of the images, however, we also need to know how the side of the images depend on the distance from the observer.  Here we can borrow a little intuition and consider the following figure with two objects at distances and :

Because of the geometric nature of optical imaging, the ratio of to is equal to the ratio of to .  In equation form, we may write

.

To put this another way, an object at distance will appear to be the same size as an object at distance if

.

This, in turn, implies that an object moved from position to position will appear to be narrower by a factor

.

If the first image a distance d has a width W, then an image at a distance 2W will appear half as wide, and an image at a distance 3W will appear one third as wide, and so on.  The same argument applies for the height H of an image at any distance.

Let’s put all this together.  The apparent area of the first image will just be ; the apparent area of the second image will be .  Continuing the process, we have the total area of all the images:

.

This is known in mathematics as an infinite series — an infinite sum of terms that may or may not add to a finite value.  It can be shown — using more math than I want to get into in this post — that the sum of this particular infinite series of terms is finite.  That is, the total area of the infinite collection of images is finite.

SO GRADUALLY FADES AWAY.