Question

Simplify (a^n+1 - b^n+1) - (a^n-1 - b^n-1) / a^n - b^n where n is even

Answer 1

Recall the proof for the sum of a geometric series:

S=1+

qS=q+q^2+…+q^{n-1}+q^n

Subtracting the second from the first:

1-q^n=(1-q)S=(1-q)(1+q+q^2+…+q^{n-1})

Now, write q=b/a and multiply by a^n, and you will obtain the identity you’re looking for.

In effect, it is the same solution as the one below by F. Farid: it is just a different, perhaps more familiar, way of thinking about it.

Answer 2

A binomial is an algebraic expression containing 2 terms. For example, (x + y) is a binomial.

We sometimes need to expand binomials as follows:

(a + b)0 = 1

(a + b)1 = a + b

(a + b)2 = a2 + 2ab + b2

(a + b)3 = a3 + 3a2b + 3ab2 + b3

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

(a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5

Clearly, doing this by direct multiplication gets quite tedious and can be rather difficult for larger powers or more complicated expressions.