Summary All Formulas Of Binomial Theorem In One Pdf

BINOMIAL THEOREM
1. STATEMENT OF BINOMIAL THEOREM
n n n n n–1 n n–2 2 n n
(x + a) = C0x + C1x a + C2 x a +..........+ Cn a (where n N)

n n n n n n!
· C0, C1 , C2,......., Cn are binomial coefficients Cr = r!(n r)!

n
General Term = Tr 1 Cr x n r a r
n
· There are (n+1) terms in the expansion of (x + a) .

· The sum of powers of a and x in each term of expansion is n.
n
· The binomial coefficients in the expansion of (x + a) equidistant from the beginning and the end are equal.

2. GREATEST BINOMIAL COFFICIENT
n n
· If n is even : When r = i.e. Cn/2 takes maximum value.
2
n 1 n 1 n
i.e. C n 1 = C n 1 and take maximum value.
n
· If n is odd : r = or
2 2 2 2



3. MIDDLE TERM OF THE EXPANSION
n n/2 n/2
If n is even T n is the middle term. So the middle term T n = Cn/2 x y
1 1
2 2



If n is odd T n 1 and T n 3 are middle terms. So the middle terms are
2 2


n 1 n 1 n 1 n 1
n n
Tn 1
C n 1
x 2
y 2
and T n 3
C n 1
x 2
y 2

2 2 2 2



4. TO DETERMINE A PARTICULAR TERM IN THE EXPANSION
n
α 1 m
In the expansion of x ± , if x occurs in Tr+1 , then r is given by
xβ
nα m
r=
α+β
nα
The term which is independent of x, occurs in Tr+1, then r is r =
α +β

5. BINOMIAL COEFFICIENT PROPERTIES
n
(1) C0 + C1 + C2 +............+ Cn = 2
n
(2) C0 – C1 + C2 – C3 +...........+ (–1) Cn = 0

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