Binomoial Expansion

(x+y)^​n​ = ⁿC₀ xⁿy⁰ + nC1 xn-1y^​1 + nC2 xn-2y^​2 + nC3 xn-3y3 + ... ... ... ... + nCn x^​0​y^​n ​

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(r+1)^th term of a binomial expansion is known as the general term. It is denoted by T_r+1​

For a binomial (a+x)ⁿ​ the general term, T_r+1 = ⁿC_r​ a^​n-r​x^​r​

Formula for finding r^th​ term, T_r​ = nCr-1​ a​n-(r-1)​x​r-1​

For a binomial (a+x)n​, 

When "n" is even it will have only one middle term that is ($\frac{n}{2}+1$)^th​ term.

When "n" is odd it will have two middle terms. They are ($\frac{n+1}{2}$)^th and ($\frac{n+3}{2}$ )th terms.

Which term is bigger 99⁵⁰ + 100⁵⁰​ = (101)⁵⁰​ ?

or, if n = 99, ​which term is bigger n50 + (n+1)50​ = (n+2)50​ ?

Solution:

(100+1)​50​ = 50C0 1005010 + 50C1 100491​1 + 50C2 100481​2 + 50C3 1004713 + ... ... ... ...  

(100-1)​50​  = 50C0 1005010 - 50C1 100491​1 + 50C2 100481​2 - 50C3 1004713 + ... ... ... ...