Prove that 1/a(a+1) = 1/a - 1/(a+1)

We notice that the denominator of the left side ratio is
the least common denominator of 2 irreducible ratios.

We'll
suppose that the ratio 1/a(a+1) is the result of addition or subtraction of 2 elementary
fractions:

1/a(a+1) = A/a + B/(a+1)
(1)

We'll multiply the ratio A/a by (a+1) and we'll
multiply the ratio B/(a+1) by a.

1/a(a+1) = [A(a+1) +
Ba]/a(a+1)

Since the denominators of both sides are
matching, we'll write the numerators, only.

1 = A(a+1) +
Ba

We'll remove the
brackets:

1 = Aa + A +
Ba

We'll factorize by a to the right
side:

1 = a(A+B) + A

If the
expressions from both sides are equivalent, the correspondent coefficients are
equal.

A+B = 0

A =
1

1 + B = 0

B =
-1

We'll substitute A and B into the expression
(1):

1/a(a+1) = 1/a -
1/(a+1)

We remark that we've obtained the
request from enunciation.

The identity has
been proved.