Greates Binomial coefficient(or Middle terms)

(1) If n is even, then there is only one middle term which (h/2 + 1) ^th term i.e.

T_n/2 + 1 = ⁿ C_n/2 X ^n/2

and ⁿC_n/2 is greatest bionomial coefficient

(2) If n is each them trere are two middle terms which are ^th and ^th terms i.e

= a ^(n+1)/2 b^(n-1)/2

and

And, are greatest bionoimial ccoefficients

Numarically Greatest term of Bionmial expansion

(a + x)ⁿ = Co aⁿ + C₁ a^n-1 x.........+ C_n-1ax^n-1 + C_nxⁿ.

The numerically greatest term will be T_r+1 where r = ....

If itself is a natural number then T_r = T_r+1

both are numerically greatest terms.

Why ?



If for given a, x and n

then

So, when

Illustration 3

Show that middle term in the expansion of (1 + x)²ⁿ is img........... where n isa +ve integer.

Ans This number of terms in expassion of (1 + x)²ⁿ is 2n + 1 (odd)

So, its middle term is (n + 1)^th term.

Required Term = T_n+1 = ²ⁿ C_n xⁿ

= xⁿ

= xⁿ

= xⁿ.

= xⁿ

=

= 2ⁿ xⁿ

Illustration 4

find the greatest term in the expansion of

Ans Let us find r =

So, r =

=

= 7

T_r+1 = T₈ is greatest term

Now T₈ = ²⁰ C₇

=

Summation of series of Bionomial co-efficients.

Series of Bionomia cofficients cn be umme by uing methods like by taaking prouct o expnion of two bionomial by differentiating bionomial expansion, integraating bionomial expansion by equating real and imaginory part of a eries etc.

(1) Sum of sevies by taking product o expnsions of two bionomials if we find the product of binomial coefficients in tnhe sevies then this method can be used .