Proving a formula with binomial coefficient when n=-1

  • Thread starter Seydlitz
  • Start date
  • #1
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Homework Statement



Prove that ##\binom{-1}{k}=(-1)^k##

The Attempt at a Solution



Using induction on ##k##,

##\binom{-1}{0}=1## which is true also for ##(-1)^0=1##

Assuming ##\binom{-1}{k}=(-1)^k##, then ##\binom{-1}{k+1}=(-1)^{k+1}##

Indeed when ##n=-1##, we can write rewrite this ##\frac{n!}{k!(n-k)!}## as ##\frac{(-1)^k(k)!}{k!}## to avoid negative factorial. Hence ##\binom{-1}{k+1}=\frac{(-1)^k(k+1)!}{(k+1)!}=(-1)^{k+1}##.

##\blacksquare##

I just want to confirm if my proof by induction method is valid.

Thank You
 

Answers and Replies

  • #3
263
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You should recast the combinatorials in terms of gamma functions before your start.

See http://mathworld.wolfram.com/BinomialCoefficient.html

This exercise is taken from Boas Mathematical Methods in Physics Chapter 1, the readers haven't been exposed to Gamma function nor does the problem actually requires one to prove the statement. It only requires one to show but I just want to prove it if possible using simple induction.
 

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