Generalisation is passing from the consideration of one
object to the consideration of a set containing that object;
or passing from the consideration of a restricted set
to that of a more comprehensive set containing the
restricted one.
1.  If, by some chance, we come across the sum

                 1 + 8 + 27 + 64 = 100

we may observe that it can be expressed in the curious
form
                1^3 + 2^3 + 3^3 + 4^3  = 10^2

Now, it is natural to ask ourselves: Does it often happen
that a sum of successive cubes as 

               1^3 + 2^3 + 3^3 + ... + n^3

is a square? In asking this, we generalise. This generalisation
is a lucky one; it leads from one observation to a 
remarkable general law. Many results were found by 
lucky generalisations in mathematics, physics, and the 
natural sciences. See INDUCTION AND MATHEMATICAL INDUCTION.
2.  Generalisation may be useful in the solution of
problems. Consider the following problem of solid geometry:
"A straight line and a rectangular octahedron are given
in position. Find a plane that passes through the given
line and bisects the volume of the given octahedron."
This problem may look difficult but, in fact, very little 
familiarity with the shape of the regular octahedron is
sufficient to suggest the following more general problem:
"A straight line and a solid with a centre of symmetry are
given in position. Find a plane that passes through
the line and bisects the volume of the given solid."
The plane required passes, of course, through the centre
of symmetry of the solid, and is determined by this point
and the given line. As the octahedron has a centre of 
symmetry, our original problem is also solved.
    The reader will not fail to observe that the second 
problem is more general than the first, and nevertheless
much easier than the first. In fact our main achievement 
in solving the first problem was to invent the second 
problem Inventing the second problem, we recognise the
role of the centre of symmetry; we disentangled that
property of the octahedron which is essential for the
problem at hand, namely that it has such a centre.
    The more general problem may be easier to solve. This
sounds paradoxical but, after the foregoing example, it 
should not be paradoxical to us. The main achievement
in solving the special problem was to invent the general 
problem. After the main achievement, only a minor part of
the work remains. Thus in our  case, the solution of
other general problem is only a minor part of the solution of
the special problem.
     See INVENTOR'S PARADOX
3.     "Find the volume of the frustum of a pyramid with
square base, being given that the side of the lower base is 
10 in, the side of the upper base 5 in, and the altitude
of the frustum 6 in." If for the numbers 10, 5, 6, we 
substitute letters, for instances, a,b,h, we generalise. We
obtain a more general problem than the original one, 
namely the following. Find the volume of the frustum
of a pyramid with square base, being given that the side 
of the lower base is a, the side of the upper base b, and 
the altitude of the frustum h."  Such generalisation may 
be very useful. Passing from a problem "in numbers" to
another one"in letters" we gain access to new procedures;
we can vary the data, and in doing so, we may check
our results in various ways. See ., 2 VARIATION OF THE PROBLEM,4.