Prove Hint Consider 1 1n and 1 1n or use Pascals equation

Prove:

Hint: Consider (1 ? 1)n and (1 + 1)n, or use Pascal’s equation and proof by induction.

Prove: Hint: Consider (1 ?? 1)n and (1 + 1)n, or use Pascal??s equation and proof by induction.

Solution

Note that

(x + y)^n = Sum [nCr x^r y^(n - r)]

Thus, if x = -1, y = 1,

[-1 + 1]^n = 0^n = 0 = Sum [nCr (-1)^r (1)^(n - r)]

0 = Sum [nCr (-1)^r (1)^(n - r)]

0 = Sum [nCr (-1)^r (1)]

0 = Sum [nCr (-1)^r] DONE!

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Note that

(x + y)^n = Sum [nCr x^r y^(n - r)]

Thus, if x = 1, y = 1,

[1 + 1]^n = 2^n = Sum [nCr 1^r * 1^(n - r)]

2^n = Sum [nCr 1 * 1]

2^n = Sum [nCr] DONE!