BINOMIAL THEOREM
INTRODUCTION
- An expression containing two terms connected by + or – sign is called a BINOMIAL. For example, a + b,
, a – y2, 2b – 3c etc. are binomial expressions.
- Similarly, an expression containing three terms is called a TRINOMIAL. In general, expressions containing more than two terms are called MULTINOMIALS.
- The general form of the binomial expression is x + a and the expansion of (x + a)n, n being a positive integer, is called the BINOMIAL THEOREM. Sir Isaac Newton first gave this theorem.
BINOMIAL THEOREM FOR POSITIVE INTEGRAL INDEX
Statement : If
, then
PROPERTIES OF THE EXPANSION OF (x + a)n
- The number of terms in the expansion of (x + a)n is n + 1, i.e., one more than the index n.
- The sum of the powers of x and a in each term is n, thus it is a homogeneous expansion.
- The (r + 1)th term in the expansion is nCr xn – r ar and is called the general term, i.e., Tr + 1 = nCr xn – r ar.
- nC0, nC1, nC2 ......, nCn are coefficients of the successive terms and are called the binomial coefficients.
- The binomial coefficients of terms equidistant from the beginning and end are same as nCr = nCn – r
- The binomial expansion is briefly written as
.
- Putting –a for a, we have
The terms in this expansion are alternatively positive and negative and the last term is positive or negative according as n is even or odd.
- Putting x = 1 and a = x in the expansion of (x + a)n, we have
This is expansion of (1 + x)n is ascending powers of x.
- Putting a = 1 in the expansion of (x + a)n, we have
This is expansion of (1 + x)n is descending powers of x. The coefficient (r + 1)th term and the coefficient of xr in (1 + x)n is or , both being equal.
- The kth term from the end in the expansion of (x + a)n is
(n – k + 2)th term from the beginning. - We note that
= 2 [sum of the terms at even places]
= 2 [sum of the term at odd places]
MIDDLE TERM OF TERMS IN THE EXPANSION OF (x + a)n
The number of terms in the expansion of (x + a)n is n + 1. Therefore,
If n is even then there is only one middle term, viz.
th term.
If n is odd then there are two middle terms, viz.
th and
th terms.
GREATEST COEFFICIENT IN THE EXPANSION OF (x + a)n
The coefficients are
where the coefficient of the general term is nCr. We have to find the value of r for which nCr has the greatest value. We know that if n is even nCr is greatest when
and if n is odd nCr is greatest for
or
.
Hence, if n is even the greatest coefficient is
, and if n is odd, the greatest coefficient is nC(n – 1)/2 or nC(n + 1)/2 both being equal.
NUMERICALLY GREATEST TERM IN THE EXPANSION OF (x + a)n
Let Tr and Tr + 1 be rth and (r + 1)th terms respectively in the binomial expansion of (x + a)n. Assume that x and a are positive
Then Tr = nCr – 1 xn – r + 1 ar – 1 and Tr + 1= nCr xn – r ar
Hence, ![](https://lh5.googleusercontent.com/_z4dOz6idt4C_p88icqqvikMbkmCPPLZMnTHiKG-kP_zKB-r-1qffPT0OHqfNyuezdadEo-O3algcFCdoJZMr21Ea7oJ2qYgm0FKa_cLb-oThQbmRMwJu5REr8xMwohN7oqoTTI)
According as
i.e., according as ![](https://lh3.googleusercontent.com/X1AvKHEGyFt9vdbF3r4_JrP41Mao56iPnEyuygJfOZhNohu_2zXFh6LK-AgNb4qzsz0db7QWDBpH7cSgrXrCGCiwIvs9JBAf7Ia5o5bP_V1AWK3hci5rwjla6OXFj6Y7cP80fbc)
i.e., according as
i.e., according as
i.e., according as
……….(I)
i.e., according as
Now the value of
may be an integer or fraction. Therefore, two cases arise.
Case I : When
is an integer, say, p
From (I), Tr + 1 > Tr if r < p, otherwise Tr + 1
Tr
∴ For r = 1, 2, ...., p – 1 we have Tr + 1 > Tr,
For r = p we have Tr + 1 = Tr
And for r = p + 1, p + 2, ..., we have Tr + 1 < Tr.
∴ Hence in this case Tp = Tp + 1 and these are greater than any other term.
Case II : When
is not an integer.
Let m be the integral part of
, i.e. ![](https://lh6.googleusercontent.com/mcJAU2k4adAYDhbmoMjQB9GoycyZ-NOzJ7GjMeFZI5MrrMtgJP2WwQS4q85jhN_PcPoRXztNd8yPhSZqjdmBWUixMjU6QESOiquXDiUcKLwJ9PQ5d3qoA6oFPnB9u60tFsMk2FI)
Then from (I), Tr + 1 > Tr for r = 1, 2, 3, ....., m and Tr + 1 < Tr for r = m + 1, m + 2, .......
Hence, in this case the Tm + 1 is the greatest term.
PROPERTIES OF BINOMIAL COEFFICIENTS
We have ![](https://lh4.googleusercontent.com/7WcB43u8eQQZlXXe0ekMHq5mnhl3kNgLleIwm_TNUWFd1O8AfRBv0aW0WU5gsYdadvoHCS-GzUPvnAt68AdFRQchvKw-SOJkdI25LqT5zfH3CVLp2UqT1xgRI1V01HJsNwVwDxU)
Also, ![](https://lh3.googleusercontent.com/J9JGsP80P3C_EKGIQ-Xs-SGGjBUgaTVg-diKdLV3-6bdTBQznwnCT8qhdYB2eS5zQiB-lXfGXgaZ2aXZ43SKVy2n_AYEAyyETWHFEfba5rV5mUBSyXrqG8_gjis6cuDrS9Vui8c)
Let us denote
by C0, C1, C2,...., Cn respectively.
Then the above expressions become ![](https://lh6.googleusercontent.com/iIpEiWrH8bZfRynQgSOiokA0kcWk3obATnMyVS5iN8JBca2g-tmFzNv8PFOnmhFCOccxj87lhNLcSyuB-7Nz79A_iHZeQfljyiJ6f7iqP3wMZAp35oCNjnMVupC5f3DX7X2t_wY)
and ![](https://lh6.googleusercontent.com/u5kIRRvXtHHnJWrZz1lpuhiebcd49iFTbUu0K2vHhlXm4CP_Q6WBr6AITgUznD8I0r7rt2GYsFMaOfUR6yxkmP9fw93hJKQCCGlLgFp2n7Bs2clgEuRJg3Ww3DlOWuX0dqPbSOI)
C0, C1, C2,....,Cn are called the binomial coefficient and have the following properties:
- In the expansion of (1 + x)n the coefficient of terms equidistant from the beginning and end are equal.
The coefficient of (r + 1)th term from the beginning is nCr. The (r + 1)th term from the end is (n – r + 1)th term from the beginning. Therefore, its coefficient is nCn – r.
But nCr = nCn – r
Hence the coefficient of terms equidistant from the beginning and end are equal.
- The sum of the binomial coefficient in the expansion of (1 + x)n is 2n.
Putting x = 1 in (1 + x)n = C0 + C1 x + C2 x2 + ..... + Cr xr + .... + Cn xn, we get
C0 + C1 + C2 + ..... + Cn = 2n or
.
- The sum of the coefficient of the odd terms is equal to the sum of the coefficient of the even terms and each is equal to
2n – 1 i.e., C0 + C2 + C4 + ...... = C1 + C3 + C5 + ..... = 2n – 1
Putting x = 1 and –1 respectively in the expansion.
we get C0 + C1 + C2 + C3 + ..... + Cn – 1 + Cn = 2n
and C0 – C1 + C2 – C3 + .... + (–1)n Cn = 0
Adding and subtracting these two equations, we get
C0 + C2 + C4 + .... = C1 + C3 + C5 + .... = 2n – 1
1 . C1 + 2 . C2 + 3 . C3 + .... + n . Cn = n . 2n – 1 or
we have ![](https://lh5.googleusercontent.com/2ZiaGqWpTekuwZ-n8_N_O4Ee_UzKv7WmtyOzhcLUN8eDPctnhnkeFrbc3iOb6ZeKmS_YRAni2qgFMYWvQwuaZkA4Qj8iJ7kw9v1_3CBF6a2f1ARvfJR2PeX0IhVfHrW5DsWb7NQ)
Differentiating both sides w.r.t. x and putting x = 1, we get ![](https://lh4.googleusercontent.com/oGVBS0S6NJrQTn5nowJMRkc0malDXnb60JYOQuM2LQFA5iCUtOQpdrqWcX-7vFwPzBXcxBJq9Ik19ZzfL6yPtaeYZTBqp1jvvmKQTFHTuhPWWehIeot1wGJiB70Jidd2N2w8hKw)
EXTRA IMPORTANT RESULTS
that is,
and