College Algebra, 2013a

684 Sequences and the Binomial Theorem
( ) ( n n
+
j − 1 j)
=
=
=
=
=
=
=
=
=
=
n!
(j − 1)!(n − (j − 1))! + n!
j!(n − j)!
n!
(j − 1)!(n − j +1)! + n!
j!(n − j)!
n!
(j − 1)!(n − j +1)(n − j)! + n!
j(j − 1)!(n − j)!
n! j
j(j − 1)!(n − j +1)(n − j)! + n!(n − j +1)
j(j − 1)!(n − j +1)(n − j)!
n! j n!(n − j +1)
+
j!(n − j +1)! j!(n − j +1)!
n! j + n!(n − j +1)
j!(n − j +1)!
n!(j +(n − j +1))
j!(n − j +1)!
(n +1)n!
j!(n +1− j))!
(n +1)!
j!((n +1)− j))!
( ) n +1
̌
j
We are now in position to state and prove the Binomial Theorem where we see that binomial
coefficients are just that - coefficients in the binomial expansion.
Theorem 9.4. Binomial Theorem: For nonzero real numbers a and b,
for all natural numbers n.
(a + b) n =
n∑
j=0
( n
j)
a n−j b j
To get a feel of what this theorem is saying and how it really isn’t as hard to remember as it may
first appear, let’s consider the specific case of n = 4. According to the theorem, we have