Solution Found!
Prove Hint: Consider (1 ? 1)n and (1 + 1)n, or use
Chapter 1, Problem 12E(choose chapter or problem)
Prove
\(\sum_{r=0}^{n}(-1)^{r}(n r)=0\) and \(\sum_{r=0}^{n}(n r)=2^{n}\)
HINT: Consider \((1-1)^{n}\) and \((1+1)^{n}\), or use Pascal's equation and proof by induction.
Equation Transcription:
Text Transcription:
Sum _r=0^n(-1^)r(nr)=0
Sum _r=0^n(nr)=2^n
(1-1)n
(1+1)n
Questions & Answers
QUESTION:
Prove
\(\sum_{r=0}^{n}(-1)^{r}(n r)=0\) and \(\sum_{r=0}^{n}(n r)=2^{n}\)
HINT: Consider \((1-1)^{n}\) and \((1+1)^{n}\), or use Pascal's equation and proof by induction.
Equation Transcription:
Text Transcription:
Sum _r=0^n(-1^)r(nr)=0
Sum _r=0^n(nr)=2^n
(1-1)n
(1+1)n
ANSWER:
Solution 12E
Step1 of 3:
We need to prove,
nCr =0 and
=
Consider (1 − 1)n and (1 + 1)n, or use Pascal’s equation and proof by induction.
Step2 of 3:
Consider,
nCr = 0
Proof: we know that the pascal’s equation is given by
nCr
Substitute a = -1 and b = 1 in above equation we get
nCr
=
nCr
0 = nCr
0 = n