b Binomial Expansion Show that for any real numbers a b we h

(b) Binomial Expansion: Show that for any real numbers a, b we have: (a b)- r=0 Conclude that (-1),(n)=0 (n)-2\" and

Solution

a)

As

(a + b)^n = Sum {C(n, r) a^r b^(n-r)}

If a = 1, b = -1,

(1+(-1))^n = Sum {C(n, r) 1^r (-1)^(n-r)}

As 1 raised to any exponent is 1, and 0 raised to any exponent is 0,

(0)^n = Sum {C(n, r) (-1)^(n-r)}

0 = Sum {C(n, r) (-1)^(n-r)}

Multiplying both sides by (-1 )^(-n),

0 = Sum {C(n, r) (-1)^(-r)}

As (-1)^(-r) = (-1)^r,

Sum {C(n, r) (-1)^r} = 0 [DONE!]

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b)

As

(a + b)^n = Sum {C(n, r) a^r b^(n-r)}

If a = 1, b = 1,

(1+1)^n = Sum {C(n, r) 1^r 1^(n-r)}

As 1 raised to any exponent is 1,

(2)^n = Sum [C(n, r)] [DONE!]

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