Binomial Theorem 8

Binomial Theorem for Positive Integral Indices

* The number of terms in an expansion is one more than the power of the binomial.
* The coefficients of the first and the last terms of an expansion are both 1.
(x + y)ⁿ=ⁿC₀xⁿ+ ⁿc₁x^{n – 1}y + ⁿC₂x^{n – 2}y²+…. + ⁿC_n – ₁xy^{n – 1} + ⁿC_nyⁿ

ⁿC₀, ⁿC₁, ⁿC₂, …ⁿC_{n – 1}, ⁿC_n are called binomial coefficients.

General and Middle Terms

In the expansion (x + y)ⁿ
* General term, Tr + 1 =ⁿC_rx^{n – r}y^r
* Middle term when n is even = ( \frac{n}{ 2}  + 1)^th term
* Middle term when n is odd =( \frac{n + 1}{ 2} )^th and  ( \frac{n + 1}{ 2}  + 1 )^th terms 

Special Cases of Binomial Theorem

(a – b)ⁿ=ⁿC₀aⁿ–ⁿc₁a^{n – 1} b + ⁿC₂a^{n – 2} b² + ….. ⁿC_n-1 a(-b)^n-1 + (-1)^{n n}C_nbⁿ
(1 + x)ⁿ = ⁿC₀ + ⁿC₁x + ⁿC₂x²+ …. ⁿC_n-1 x^n-1 + ⁿC_nxⁿ
(1 + y)ⁿ=ⁿC₀ –ⁿC₁ y + ⁿC₂y²–ⁿC₃ y³+ …. + (-1)^{n n}C_n(y)^n
nC₀ + ⁿc₁ + ⁿC₂ + …. ⁿC_n-1 + ⁿC_n = 2^n
nC₀ + ⁿC₂ + ⁿC₄ + …. = ⁿC₁ + ⁿC₃ + ⁿC₅….