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Blast into Math! Mathematical perspectives: all aour mase are melong to us
Next we can bring the a and b inside the sum, and the exponent rules tell us that
n n! n n!
j n−j+1
b
(a + b) n+1 = a k+1 n−k + a b
k!(n − k)! j!(n − j)!
k=0 j=0
Let’s calmly and carefully look at this. Our goal is to reach a formula with a sum from 0 to
1
n +1, but these sums go from 0 to n . The first sum has powers of a from a up to a n+1 ,
a
and the second sum has powers of from a up to a . Since the terms in the first sum have
n
0
0
1
powers from a up to a n+1 , the only term with a is from the second sum. This term is the
j =0 term:
n!
0 n+1
a b = b n+1 .
0!n!
This term is the same as the term with k =0 in:
n+1
k n−k (n +1)!
a b .
k!(n +1 − k)!
k=0
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