Engineering Mathematics: Binomial theorem

Monday, August 1, 2011

BT.ID.1: Statement of the theorem

$(x+y)^{n}= \sum _{k=0}^{n}\binom{n}{k}x^{n-k}y^{k}= \sum _{k=0}^{n}\binom{n}{k}x^{k}y^{n-k}$

(1)

$(1+x)^{n}= \sum _{k=0}^{n}\binom{n}{k}x^{k}$

(2)

if x = 0 , then

$\sum _{k=0}^{n}\binom{n}{k}=2^{n}$

(3)

BT.ID.2: Trigonometric Identity

if n is replaced by 2n in (3) , then:

$\sum _{k=0}^{2n}\binom{2n}{k}=2^{2n}$

(4)

breaking down the left hand side

$\sum _{k=0}^{n-1}\binom{2n}{k} + \binom{2n}{k} + \sum _{k=n+1}^{2n}\binom{2n}{k}=2^{2n}$

(5)

$\sum _{k=0}^{n-1}\binom{2n}{k} + \binom{2n}{k} + \sum _{k=0}^{n-1}\binom{2n}{2n-k}=2^{2n}$

(6)

$\sum _{k=0}^{n-1}\binom{2n}{k} + \binom{2n}{k} + \sum _{k=0}^{n-1}\binom{2n}{k}=2^{2n}$

(7)

$\binom{2n}{k} =2^{2n} - 2\sum _{k=0}^{n-1}\binom{2n}{k}$

Engineering Mathematics