Mathematical proof of charging by induction If we bring a positive charge  +Q near a neutral conductor , we know that the surface near the source gets-Q and opposite  to it gets +Q, but why do these induced charges have to be equal in magnitude to source charge, why isn't a charge distribution such as -7Q on surface near the source and +7Q on the opposite surface, not possible ? Can we show the result mathematically?
 A: In the equilibrium situation any free charge inside the conductor has no prefered direction to travel.
If a charge Q is near the conductor, and 7Q of opposing changes at its surface, clearly a free charge would have a prefered direction.
As the are plenty of free charges in a conductor, eventually (and very quick indeed) the charges redistribute in order to reach the equilibrium.
A: Charges are free to flow inside a conductor. If there is an electric field inside a conductor, positive charges will flow with the field and negative charges against it. These displaced charges contribute their own electric field that quickly cancels out the original field. This is why we say there the electric field is always zero inside a conductor.
Now, getting back to your question. If a change of +Q induced a charge of -7Q on the surface nearest, then the overall charge in that area would be very negative. Charges of the same sign repel, so the accumulated charges would repel each other and the induced charge in the conductor would decrease.
If you want a more mathematical answer, take a look at the explicit examples in this article: https://en.wikipedia.org/wiki/Method_of_image_charges
