Define the binomial coefficient by = For r=1,2,3,…..,n. (a) Show that + = for r=1,2,3,….,n. (b) Use part (a) and mathematical induction to prove the binomial theorem: (a+b)n+ an + an-1b +…..+...

Define the binomial coefficient

by

=

For r=1,2,3,…..,n.

(a) Show that

+

=

for r=1,2,3,….,n.

(b) Use part (a) and mathematical induction to prove the binomial theorem:

(a+b)ⁿ+
aⁿ
+
a^n-1b +…..+
a^n-rb^r+…..+bⁿ
= aⁿ+na^n-1b+n(n-1)a^n-2b²
+…….+ nab^n-1+bⁿ