Bionomial Theoram

Bionomial Theoram

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Bionomial Theoram

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 ?

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
=

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 .

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