StudySteps.in Binomial Theorem 1. DEFINITION OF BINOMIAL EXPRESSIONAND BINOMIAL EXPANSION : An expression containing two terms, is called a binomial expression. For example a + b/x, x + 1/y, a – y 2 etc. are binomial expressions. Expansion of (x + a) n is called Binomial Expansion. Expression containing three terms are called trinomials. For example x + y + z is a trinomial expression. In general an expression containing more than two terms is called a multinomial. 1.1 Definition of binomial theorem : If n is a positive integer and x, y are two complex numbers, then n n n nr r r r0 x y Cx y = n C 0 x n + n C 1 x n – 1 y + n C 2 x n – 2 y 2 + . . . + n C n y n . . . (i) The coefficients n C 0 , n C 1 , . . . , n C n are called binomial coefficients, while (i) is called the binomial expansion. 1.2 Some Important Facts Regarding Binomial Expansion : (i) There are (n + 1) terms in the expansion. (ii) The sum of the exponents of x and y in any term of the expansion is equal to n. (iii) The binomial coefficients of terms equidistant from the beginning and the end are equal, since n C r = n C n – r . (iv) The term n C r x n – r y r is the (r + 1)th term from the beginning of the expansion. It is usually denoted by T r + 1 and is called the general term of the expansion. (v) The rth term from the end is equal to the (n – r + 2)th term from the beginning, i.e., n C n – r + 1 x r – 1 y n – r + 1 . (vi) If n is even, then the expansion has only one middle term, the
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Binomial Theorem - WordPress.com...ein Binomial Theorem i.e., r = 6. Thus, the 7th term has x– 8 and its coefficient is 10 6 6 6 105 C 1 2 32 . Illustration 2: Find the 3rd term
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Study
Steps
.in
Binomial Theorem
1. DEFINITION OF BINOMIAL EXPRESSIONAND BINOMIAL EXPANSION :An expression containing two terms, is called a binomial expression. For examplea + b/x, x + 1/y, a – y2 etc. are binomial expressions. Expansion of (x + a)n is called BinomialExpansion.Expression containing three terms are called trinomials. For example x + y + z is a trinomial expression.In general an expression containing more than two terms is called a multinomial.
1.1 Definition of binomial theorem :If n is a positive integer and x, y are two complex numbers, then
n
n n n r rr
r 0
x y C x y
= nC0xn + nC
1xn – 1 y + nC
2xn – 2 y2 + . . . + nC
n yn . . . (i)
The coefficients nC0, nC
1, . . . , nC
n are called binomial coefficients, while (i) is called the binomial
expansion.
1.2 Some Important Facts Regarding Binomial Expansion :
(i) There are (n + 1) terms in the expansion.
(ii) The sum of the exponents of x and y in any term of the expansion is equal to n.
(iii) The binomial coefficients of terms equidistant from the beginning and the end are equal,since nC
r = nCn – r .
(iv) The term nCr xn – r yr is the (r + 1)th term from the beginning of the expansion. It is usually
denoted by Tr + 1
and is called the general term of the expansion.
(v) The rth term from the end is equal to the (n – r + 2)th term from the beginning, i.e.,nCn – r + 1
xr – 1 yn – r + 1 .
(vi) If n is even, then the expansion has only one middle term, then
12
th term i.e.,
n n / 2 n / 2n / 2C x y .
If n is odd, then the expansion has two middle terms, then 1
2
th term and then 3
2
th
term i.e., n 1 / 2 n 1 / 2n
n 1 / 2C x y and
n 1 / 2 n 1 / 2nn 1 / 2C x y .
Illustration 1 :
Find the coefficient of x–8 in the expansion of10
2
1x
2x
.
Solution:We have
Tr + 1
= 10Cr(x)10 – r
r
2
1
2x
= 10Cr (– 1)r 2– r x10 – 3r
To find the coefficient of x–8, we have10 – 3r = – 8
5. If (5 + 6 )n = I + f, where I and n are positive integers and f is a positive fraction less than one, show
that (I + f) (1 – f) = 1.
3. GREATEST BINOMIAL COEFFICIENT :The greatest coefficient depends upon the value of n. n no. of greatest coefficient (s) Greatest coefficientEven 1 nC
n/2
Odd 2 n 1
n
2
C and n 1
n
2
C
(Values of both these coefficients are equal)Clearly greatest binomial coefficient corresponds to the coefficient of middle term.
4. NUMERICALLY GREATEST TERM OF BINOMIAL EXPANSION :(a + x)n = C
0an + C
1an – 1 x + . . . C
n – 1 a xn – 1 + Cnxn
nr 1 r
nr r 1
T C x n r 1 x
T a r aC
Ifn r 1 x
1r a
, for given a, x and n, then r n 1
a1
x
So numerically greatest term will be Tr + 1
, where r =n 1
a1
x
[ ] denotes the greatest integer function.
Note : Ifn 1
a1
x
itself is a natural number, then TT
r = T
r + 1 and both the terms are numerically greatest term.
a a a a n a n i kk i1 2 3 0 1 2 3 .......... , , , , ,....... and the number of terms in the expansion aren k
kC
11.
Number of terms in (x + y)n = n C1 1
Number of terms in ( )x y z Cn n 22
Number of terms in ( )x y z Cn 33
5.5 Sum of the series by comparing the coefficients of some power of x in an expansion :In this method we use the fact that coefficient of same power of x in an appropriate identity is the givenseries.
Important Formulae :
If C C C C0 1 2 3, , ,..........., represent n n n nC C C C0 1 2 3, , ..........., in the expansion of (1 + x)n . Then
(i) C C C C Cnn
n02
12
22 2 2 .............
(ii) C C C C C C C C C or Cr r r n r nn
n rn
n r0 1 1 2 22 2 .............
Illustration 10:Find the sum of the series
mCr + mCr – 1
nC1+ mCr – 2
nC2 + . . . + nC
r
where r < m, n and m, n, r N.
Solution :We have
(1 + x)n = nC0 + nC
1 x + nC
2x2 + . . . + nC
rxr + . . . + nC
n xn . . . (ii)
and (1 + x)m = mC0 + mC
1x + . . . + mCr – 2 xr – 2 + mCr – 1 x
r – 1
+ mCr xr + . . . + mC
mxm . . . (ii)
Hence, the given series= coefficient of xr in (1 + x)n (1 + x)m
= coefficient of xr in (1 + x)m + n = m nrC .
Illustration 11:
Find the sum of the series 2 2 2 21 2 3 nC 2.C 3.C . . . n.C
If x < 1, the terms of the above expansion go on decreasing and if x be very small, a stage may bereached when we may neglect the terms containing higher powers of x in the expansion. Thus, ifx be so small that its squares and higher powers may be neglected then (1 + x)n = 1 + nx,approximately . This is an approximate value of (1 + x)n .
5.8 Exponential Series :
(i) ex = .........!3
x
!2
x
!1
x1
32
;where x may be any real or complex and e =n
n n
11Lim
(ii) ax = .......an!3
xan
!2
xna
!1
x1 3
32
2
where a > 0
Note : (a) e = .......!3
1
!2
1
!1
11
(b) e is an irrational number lying between 2.7 and 2.8. Its value correct upto 10 place ofdecimal is 2.7182818284.
(c) e + e-1 =
........
!6
1
!4
1
!2
112
(d) e - e-1 =
........
!7
1
!5
1
!3
112
(e) Logarithms to the base ‘e’ are known as the Napierian system, so named after Napier,their inventor. They are also called Natural Logarithm.